{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":44724,"title":"Let's Make A Deal: The Player's Dilemma 2","description":"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, either immediately opens your door or opens another door, say No. 3, which might have the car or a goat. If the host opens door No. 3 and it has a goat behind it, he then offers you the choice, \"Do you want to pick door No. 2?\" Is it to your advantage to switch your choice?\r\n\r\nWe will now play the game using the assumptions:\r\n\r\n# The host can open any of the doors including that picked by the player.\r\n# The host can open a door to reveal either a goat or the car.\r\n# The host doesn't always offer the chance to switch between the originally chosen \r\n  door and the closed doors.\r\n\r\nIt is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door D, then the host's choice of which door to open is represented by a conditional probability matrix Ph\r\n\r\n   Ph = [ p_11   p_12   p_13\r\n          p_21   p_22   p_23\r\n          p_31   p_32   p_33 ]\r\n\r\nIn the above matrix, p_ij represents the probability that the host opens door j given that the car is behind door i.\r\n\r\nInterpreting the matrix in terms of the assumptions implies that all elements p_ij of the conditional probability matrix can be nonzero with the constraint that\r\n\r\n  p_i1 + p_i2 + p_i3 = 1    i.e. the host must always open a door to reveal the car or a goat. \r\n\r\nOn the game show, you have initially chosen door D and the host, Monty Hall, opened door H, using the conditional probability Ph.\r\n\r\nWhat is the probability *Pws* that you will win the car by switching your choice to the door remaining?\r\n\r\n*If switching option is not available (i.e. host opens a door with the car behind it) return NaN.*\r\n","description_html":"\u003cp\u003eSuppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, either immediately opens your door or opens another door, say No. 3, which might have the car or a goat. If the host opens door No. 3 and it has a goat behind it, he then offers you the choice, \"Do you want to pick door No. 2?\" Is it to your advantage to switch your choice?\u003c/p\u003e\u003cp\u003eWe will now play the game using the assumptions:\u003c/p\u003e\u003col\u003e\u003cli\u003eThe host can open any of the doors including that picked by the player.\u003c/li\u003e\u003cli\u003eThe host can open a door to reveal either a goat or the car.\u003c/li\u003e\u003cli\u003eThe host doesn't always offer the chance to switch between the originally chosen \r\n  door and the closed doors.\u003c/li\u003e\u003c/ol\u003e\u003cp\u003eIt is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door D, then the host's choice of which door to open is represented by a conditional probability matrix Ph\u003c/p\u003e\u003cpre\u003e   Ph = [ p_11   p_12   p_13\r\n          p_21   p_22   p_23\r\n          p_31   p_32   p_33 ]\u003c/pre\u003e\u003cp\u003eIn the above matrix, p_ij represents the probability that the host opens door j given that the car is behind door i.\u003c/p\u003e\u003cp\u003eInterpreting the matrix in terms of the assumptions implies that all elements p_ij of the conditional probability matrix can be nonzero with the constraint that\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003ep_i1 + p_i2 + p_i3 = 1    i.e. the host must always open a door to reveal the car or a goat. \r\n\u003c/pre\u003e\u003cp\u003eOn the game show, you have initially chosen door D and the host, Monty Hall, opened door H, using the conditional probability Ph.\u003c/p\u003e\u003cp\u003eWhat is the probability \u003cb\u003ePws\u003c/b\u003e that you will win the car by switching your choice to the door remaining?\u003c/p\u003e\u003cp\u003e\u003cb\u003eIf switching option is not available (i.e. host opens a door with the car behind it) return NaN.\u003c/b\u003e\u003c/p\u003e","function_template":"function Pws = MontyHall2(D,H,Ph)\r\n  Pws = ;\r\nend","test_suite":"\r\n%% CASE 1: Host always opens the door containing the car.\r\n%      Ph = [1   0   0\r\n%            0   1   0\r\n%            0   0   1]\r\n% In this case, the host does not offer the player the choice of switching\r\n% doors. Hence, return Pws = NaN.\r\n%__________________________________________________________________________\r\nD = 1;\r\nH = 2;\r\nPh = diag([1 1 1]);\r\nPws_correct = NaN;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 1;\r\nH = 3;\r\nPh = diag([1 1 1]);\r\nPws_correct = NaN;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = diag([1 1 1]);\r\nPws_correct = NaN;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n\r\n%% CASE 2: Host (uniformly) randomly opens any door.\r\n%      Ph = [1/3  1/3  1/3\r\n%            1/3  1/3  1/3\r\n%            1/3  1/3  1/3]\r\n% In this case, the game is fair because the host does not make any effort\r\n% to open any particular door. Hence, the chance of winning or losing by\r\n% switching is equal i.e. Psw = 0.5.\r\n%__________________________________________________________________________\r\n\r\nD = 1;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPws_correct = 0.5;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 1;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPws_correct = 0.5;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 1;\r\nPh = ones(3)/3;\r\nPws_correct = 0.5;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPws_correct = 0.5;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%% CASE 3: Host opens door 2 if the car is not behind it. Otherwise,\r\n% the host opens door 3.\r\n%      Ph = [0   1   0\r\n%            0   0   1\r\n%            0   1   0]\r\n%__________________________________________________________________________\r\n\r\nD = 1;\r\nH = 3;\r\nPh = [0 1 0;\r\n      0 0 1;\r\n      0 1 0];\r\nPws_correct = 1;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 1;\r\nH = 2;\r\nPh = [0 1 0;\r\n      0 0 1;\r\n      0 1 0];\r\nPws_correct = 0.5;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0 1 0;\r\n      0 0 1;\r\n      0 1 0];\r\nPws_correct = 0;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [0 1 0;\r\n      0 0 1;\r\n      0 1 0];\r\nPws_correct = 0.5;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%% CASE 4: Host opens the door based on the standard assumptions of \r\n%  Marilyn vos Savant.\r\n%__________________________________________________________________________\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPws_correct = 0.6667;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPws_correct = 0.6667;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPws_correct = 0.3333;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%% CASE 5: Host opens the door based on a completely random approach.\r\n%__________________________________________________________________________\r\n\r\nD = 1;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPws_correct = 0.5045;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPws_correct = 0.4955;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPws_correct = 0.4745;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [0.5000    0.1000    0.4000\r\n      0.0500    0.5000    0.4500\r\n      0.3333    0.0667    0.6000];\r\nPws_correct = 0.8696;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0.5000    0.1000    0.4000\r\n      0.0500    0.5000    0.4500\r\n      0.3333    0.0667    0.6000];\r\nPws_correct = 0.4706;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [0.5000    0.1000    0.4000\r\n      0.0500    0.5000    0.4500\r\n      0.3333    0.0667    0.6000];\r\nPws_correct = 0.1304;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":178544,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2018-08-10T08:50:31.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2018-08-08T10:20:24.000Z","updated_at":"2018-08-10T08:50:31.000Z","published_at":"2018-08-08T12:55:34.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSuppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, either immediately opens your door or opens another door, say No. 3, which might have the car or a goat. If the host opens door No. 3 and it has a goat behind it, he then offers you the choice, \\\"Do you want to pick door No. 2?\\\" Is it to your advantage to switch your choice?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWe will now play the game using the assumptions:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host can open any of the doors including that picked by the player.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host can open a door to reveal either a goat or the car.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host doesn't always offer the chance to switch between the originally chosen door and the closed doors.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door D, then the host's choice of which door to open is represented by a conditional probability matrix Ph\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   Ph = [ p_11   p_12   p_13\\n          p_21   p_22   p_23\\n          p_31   p_32   p_33 ]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the above matrix, p_ij represents the probability that the host opens door j given that the car is behind door i.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInterpreting the matrix in terms of the assumptions implies that all elements p_ij of the conditional probability matrix can be nonzero with the constraint that\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[p_i1 + p_i2 + p_i3 = 1    i.e. the host must always open a door to reveal the car or a goat.]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the game show, you have initially chosen door D and the host, Monty Hall, opened door H, using the conditional probability Ph.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat is the probability\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ePws\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e that you will win the car by switching your choice to the door remaining?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eIf switching option is not available (i.e. host opens a door with the car behind it) return NaN.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44727,"title":"Let's Make A Deal: Criminal Minds","description":"The game of Let's Make A Deal proceeds as follows;\r\n\r\nThe player is given the choice of three doors: Behind one door is a car; behind the others, goats. The player picks a door, say No. 1, and the host, either immediately opens that door or opens another door, say No. 3, which might have the car or a goat. If the host opens door No. 3 and it has a goat behind it, he then offers the player the choice, \"Do you want to pick door No. 2?\"\r\n\r\nThe game play is guided by the following set of rules:\r\n\r\n# The host can open any of the doors including that picked by the player.\r\n# The host can open a door to reveal either a goat or the car.\r\n# The host doesn't always offer the chance to switch between the originally chosen door and the closed doors.\r\n\r\nIn order to manipulate the game play, the host does not always (uniformly) randomly hide the car behind the doors. Instead, based on previous games statistics, the host determines the probability Pc. \r\n\r\n Pc = [pc_1  pc_2  pc_3]   pc_i is the probability of the host placing car behind door i. \r\n\r\nIf the player initially picks door D, then the host's choice of which door to open is represented by a conditional probability matrix Ph\r\n\r\n   Ph = [ ph_11   ph_12   ph_13\r\n          ph_21   ph_22   ph_23\r\n          ph_31   ph_32   ph_33 ]\r\n\r\nIn the above matrix, ph_ij represents the probability that the host opens door j given that the car is behind door i.\r\n\r\nInterpreting the matrix in terms of the assumptions implies that all elements ph_ij of the conditional probability matrix can be nonzero with the constraint that\r\n\r\n ph_i1 + ph_i2 + ph_i3 = 1    i.e. the host must always open a door to reveal the car or a goat. \r\n\r\nOn the game show, the host places the car behind a door with probability Pc in order to manipulate the outcome. The player then choose door D and the host, Monty Hall, opens door H using the conditional probability Ph.\r\n\r\nIf the door H opened by the host has a goat behind it, what is the probability *Pws* that the player will win the car by switching the initial door choice to the door remaining?\r\n\r\n*If switching option is not available (i.e. host always opens a door with the car behind it) return NaN.*\r\n","description_html":"\u003cp\u003eThe game of Let's Make A Deal proceeds as follows;\u003c/p\u003e\u003cp\u003eThe player is given the choice of three doors: Behind one door is a car; behind the others, goats. The player picks a door, say No. 1, and the host, either immediately opens that door or opens another door, say No. 3, which might have the car or a goat. If the host opens door No. 3 and it has a goat behind it, he then offers the player the choice, \"Do you want to pick door No. 2?\"\u003c/p\u003e\u003cp\u003eThe game play is guided by the following set of rules:\u003c/p\u003e\u003col\u003e\u003cli\u003eThe host can open any of the doors including that picked by the player.\u003c/li\u003e\u003cli\u003eThe host can open a door to reveal either a goat or the car.\u003c/li\u003e\u003cli\u003eThe host doesn't always offer the chance to switch between the originally chosen door and the closed doors.\u003c/li\u003e\u003c/ol\u003e\u003cp\u003eIn order to manipulate the game play, the host does not always (uniformly) randomly hide the car behind the doors. Instead, based on previous games statistics, the host determines the probability Pc.\u003c/p\u003e\u003cpre\u003e Pc = [pc_1  pc_2  pc_3]   pc_i is the probability of the host placing car behind door i. \u003c/pre\u003e\u003cp\u003eIf the player initially picks door D, then the host's choice of which door to open is represented by a conditional probability matrix Ph\u003c/p\u003e\u003cpre\u003e   Ph = [ ph_11   ph_12   ph_13\r\n          ph_21   ph_22   ph_23\r\n          ph_31   ph_32   ph_33 ]\u003c/pre\u003e\u003cp\u003eIn the above matrix, ph_ij represents the probability that the host opens door j given that the car is behind door i.\u003c/p\u003e\u003cp\u003eInterpreting the matrix in terms of the assumptions implies that all elements ph_ij of the conditional probability matrix can be nonzero with the constraint that\u003c/p\u003e\u003cpre\u003e ph_i1 + ph_i2 + ph_i3 = 1    i.e. the host must always open a door to reveal the car or a goat. \u003c/pre\u003e\u003cp\u003eOn the game show, the host places the car behind a door with probability Pc in order to manipulate the outcome. The player then choose door D and the host, Monty Hall, opens door H using the conditional probability Ph.\u003c/p\u003e\u003cp\u003eIf the door H opened by the host has a goat behind it, what is the probability \u003cb\u003ePws\u003c/b\u003e that the player will win the car by switching the initial door choice to the door remaining?\u003c/p\u003e\u003cp\u003e\u003cb\u003eIf switching option is not available (i.e. host always opens a door with the car behind it) return NaN.\u003c/b\u003e\u003c/p\u003e","function_template":"function Pws = MontyHall3(D,H,Ph,Pc)\r\n  \r\nend","test_suite":"%% CASE 1: Monty Hall (uniformly) randomly opens any door.\r\n%      Ph = [1/3  1/3  1/3\r\n%            1/3  1/3  1/3\r\n%            1/3  1/3  1/3]\r\n%__________________________________________________________________________\r\n%\r\n% CASE 1A: Host always hides the car behind one particular door.\r\n%   Pc = [1  0  0] or [0  1  0]  or  [0  0  1]\r\n%__________________________________________________________________________\r\n% 1. Pc = [1 0 0], D = 1,  H = 3. \r\n% The host always hides the car behind door 1.\r\n% Consequently, a player with an initial door choice of D = 1 cannot win\r\n% by switching doors after the host opens door H = 3 i.e. Psw = 0.\r\nD = 1;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPc = [1  0  0];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n% 2. Pc = [1 0 0], D = 1, H = 2.\r\n% The host always hides the car behind door 1.\r\n% Consequently, a player with an initial door choice of D = 1 cannot win\r\n% by switching doors after the host opens door H = 2 i.e. Psw = 0.\r\nD = 1;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [1  0  0];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n% 3. Pc = [0 1 0], D = 1, H = 2. \r\n% The host always hides the car behind door 2.\r\n% Consequently, a player with an initial door choice of D = 1 does not\r\n% have the option to switch doors after the host opens door H = 2.\r\nD = 1;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [0  1  0];\r\nPws_correct = NaN;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n% 4. Pc = [0 1 0], D = 1, H = 3. \r\n% The host always hides the car behind door 2.\r\n% Consequently, a player with an initial door choice of D = 1 can only win\r\n% by switching to door 2 after the host opens door H = 3. i.e. Psw = 1.\r\nD = 1;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPc = [0  1  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n% 5. Pc = [0 0 1], D = 1, H = 2\r\n% The host always hides the car behind door 3.\r\n% Consequently, a player with an initial door choice of D = 1 does not\r\n% have the option to switch doors after the host opens door H = 2.\r\nD = 1;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [0  0  1];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n\r\n% 6. Pc = [0 0 1], D = 1, H = 3\r\n% The host always hides the car behind door 3.\r\n% Consequently, a player with an initial door choice of D = 1 does not\r\n% have the option to switch doors after the host opens door H = 3.\r\nD = 1;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPc = [0  0  1];\r\nPws_correct = NaN;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 1B: Host hides the car behind two particular doors with equal probability.\r\n% Pc = [0  0.5  0.5]  or [0.5  0.5  0]  or  [0.5  0  0.5]\r\n%__________________________________________________________________________\r\n% 1. Pc = [0  0.5  0.5], D = 3, H = 1.\r\n% The car is either behind door 2 or 3. If the host opens door H = 1, the\r\n% player's chance of winning by switching from door D = 3 to door 2 is\r\n% equal i.e. Psw = 0.5.\r\nD = 3;\r\nH = 1;\r\nPh = ones(3)/3;\r\nPc = [0  0.5  0.5];\r\nPws_correct = 0.5;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n% 2. Pc = [0  0.5  0.5], D = 3, H = 2.\r\n% The host never hides the car behind door 1. Hence, the player can never\r\n% wi by switching to door 1.\r\nD = 3;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [0  0.5  0.5];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n% 3. Pc = [0.5  0.5  0], D = 3, H = 1.\r\n% The host never hides the car behind door 3. Hence, the player can only \r\n% win by switching to door 2.\r\nD = 3;\r\nH = 1;\r\nPh = ones(3)/3;\r\nPc = [0.5  0.5  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n% 4. Pc = [0.5  0.5  0], D = 3, H = 2.\r\n% The host never hides the car behind door 3. Hence, the player can only \r\n% win by switching to door 1.\r\nD = 3;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [0.5  0.5  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n% 5. Pc = [0.5  0  0.5], D = 3, H = 1.  \r\n% The host never hides the car behind door 2. Hence, the player can\r\n% never win by switching to door 2.\r\nD = 3;\r\nH = 1;\r\nPh = ones(3)/3;\r\nPc = [0.5  0  0.5];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 1C: Hosts hides the car behind any of the three doors with equal probability.\r\n%   Pc = [1/3  1/3  1/3]\r\n%__________________________________________________________________________\r\n% The host does not manipulate the game. Hence, it is a fair game.\r\nD = 1;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPc = [1  1  1]/3;\r\nPws_correct = 0.5;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 1D: Host hides the car with a pre-determined probability.\r\n%   Pc = [0.26  0.43  0.31]\r\n%__________________________________________________________________________\r\n\r\nD = 1;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.6232;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 1;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.5439;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 1;\r\nPh = ones(3)/3;\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.4189;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.3768;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = ones(3)/3;\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.5811;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.4561;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%% CASE 2: Monty Hall opens the door based on the standard assumptions of \r\n%  Marilyn vos Savant.\r\n%      Ph = [ 0    1   0\r\n%             1    0   0\r\n%            0.5  0.5  0]\r\n%__________________________________________________________________________\r\n%\r\n% CASE 2A: Host always hides the car behind one particular door.\r\n%   Pc = [1  0  0] or [0  1  0]  or  [0  0  1]\r\n%__________________________________________________________________________\r\n \r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [1  0  0];\r\nPws_correct = NaN;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [1  0  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0  1  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0  1  0];\r\nPws_correct = NaN;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0  0  1];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0  0  1];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 2B: Host hides the car behind two particular doors with equal \r\n% probability.\r\n% Pc = [0  0.5  0.5]  or [0.5  0.5  0]  or  [0.5  0  0.5]\r\n%__________________________________________________________________________\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.5  0.5  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.5  0.5  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0  0.5  0.5];\r\nPws_correct = 0.6667;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0  0.5  0.5];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.5  0  0.5];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.5  0  0.5];\r\nPws_correct = 0.6667;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 2C: Hosts hides the car behind any of the three doors with equal \r\n% probability.\r\n%   Pc = [1/3  1/3  1/3]\r\n%__________________________________________________________________________\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [1  1  1]/3;\r\nPws_correct = 0.6667;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [1  1  1]/3;\r\nPws_correct = 0.6667;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 2D: Host hides the car with a pre-determined probability.\r\n%   Pc = [0.26  0.43  0.31]\r\n%__________________________________________________________________________\r\n\r\nD = 1;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.3735;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.2650;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.7350;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.6265;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n\r\n%% CASE 3: Host opens the door based on a completely randomized approach.\r\n%__________________________________________________________________________\r\n%\r\n% CASE 3A: Host always hides the car behind one particular door.\r\n%   Pc = [1  0  0] or [0  1  0]  or  [0  0  1]\r\n%__________________________________________________________________________\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [1  0  0];\r\nPws_correct = NaN;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [1  0  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0  1  0];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0  0  1];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0  0  1];\r\nPws_correct = NaN;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 3B: Host hides the car behind two particular doors with equal probability.\r\n% Pc = [0  0.5  0.5]  or [0.5  0.5  0]  or  [0.5  0  0.5]\r\n%__________________________________________________________________________\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0  0.5  0.5];\r\nPws_correct = 0.6136;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0  0.5  0.5];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.5  0.5  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.5  0.5  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.5  0  0.5];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 3C: Hosts hides the car behind any of the three doors with equal \r\n% probability.\r\n%   Pc = [1/3  1/3  1/3]\r\n%__________________________________________________________________________\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [1  1  1]/3;\r\nPws_correct = 0.6136;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [1  1  1]/3;\r\nPws_correct = 0.4745;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 3D: Host hides the car with a pre-determined probability.\r\n%   Pc = [0.26  0.43  0.31]\r\n%__________________________________________________________________________\r\n\r\nD = 1;\r\nH = 2;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.5690;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 1;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.6274;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.3123;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.3726;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.6877;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.4310;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":1,"created_by":178544,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2018-08-10T13:05:15.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-08-09T10:04:50.000Z","updated_at":"2018-08-10T13:05:15.000Z","published_at":"2018-08-10T13:00:57.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe game of Let's Make A Deal proceeds as follows;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe player is given the choice of three doors: Behind one door is a car; behind the others, goats. The player picks a door, say No. 1, and the host, either immediately opens that door or opens another door, say No. 3, which might have the car or a goat. If the host opens door No. 3 and it has a goat behind it, he then offers the player the choice, \\\"Do you want to pick door No. 2?\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe game play is guided by the following set of rules:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host can open any of the doors including that picked by the player.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host can open a door to reveal either a goat or the car.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host doesn't always offer the chance to switch between the originally chosen door and the closed doors.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn order to manipulate the game play, the host does not always (uniformly) randomly hide the car behind the doors. Instead, based on previous games statistics, the host determines the probability Pc.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Pc = [pc_1  pc_2  pc_3]   pc_i is the probability of the host placing car behind door i.]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf the player initially picks door D, then the host's choice of which door to open is represented by a conditional probability matrix Ph\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   Ph = [ ph_11   ph_12   ph_13\\n          ph_21   ph_22   ph_23\\n          ph_31   ph_32   ph_33 ]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the above matrix, ph_ij represents the probability that the host opens door j given that the car is behind door i.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInterpreting the matrix in terms of the assumptions implies that all elements ph_ij of the conditional probability matrix can be nonzero with the constraint that\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ ph_i1 + ph_i2 + ph_i3 = 1    i.e. the host must always open a door to reveal the car or a goat.]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the game show, the host places the car behind a door with probability Pc in order to manipulate the outcome. The player then choose door D and the host, Monty Hall, opens door H using the conditional probability Ph.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf the door H opened by the host has a goat behind it, what is the probability\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ePws\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e that the player will win the car by switching the initial door choice to the door remaining?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eIf switching option is not available (i.e. host always opens a door with the car behind it) return NaN.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44723,"title":"Let's Make A Deal: The Player's Dilemma ","description":"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, \"Do you want to pick door No. 2?\" Is it to your advantage to switch your choice?\r\n\r\nWe will now play the game using the standard assumptions:\r\n\r\n# The host must always open a door that was not picked by the contestant.\r\n# The host must always open a door to reveal a goat and never the car.\r\n# The host must always offer the chance to switch between the originally chosen door and the remaining closed door.\r\n\r\nIt is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door 1, then the host's choice of which goat-hiding door to open is represented by a conditional probability matrix Ph \r\n     \r\n   Ph = [ p_11   p_12   p_13\r\n          p_21   p_22   p_23\r\n          p_31   p_32   p_33 ]\r\n\r\nIn the above matrix, p_ij represents the probability that the host opens door j\r\ngiven that the car is behind door i. \r\n\r\nInterpreting the matrix in terms of the standard assumptions implies \r\n\r\n p_i1 = 0            i.e.  the host cannot open door 1, the player's initial choice.  \r\n  \r\n p_i2 + pi3 = 1      i.e. the host must always open a door, 2 or 3, not initially picked by the player.\r\n\r\n p_ii = 0            i.e. the host must always open a door to reveal a goat and never the car.\r\n\r\nOn the game show, you have initially chosen door 1 and the host, Monty Hall, opened door H (2 or 3), using the conditional probability Ph. \r\n\r\nWhat is the probability *Pws* that you will win the car by switching your choice to the door remaining? ","description_html":"\u003cp\u003eSuppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, \"Do you want to pick door No. 2?\" Is it to your advantage to switch your choice?\u003c/p\u003e\u003cp\u003eWe will now play the game using the standard assumptions:\u003c/p\u003e\u003col\u003e\u003cli\u003eThe host must always open a door that was not picked by the contestant.\u003c/li\u003e\u003cli\u003eThe host must always open a door to reveal a goat and never the car.\u003c/li\u003e\u003cli\u003eThe host must always offer the chance to switch between the originally chosen door and the remaining closed door.\u003c/li\u003e\u003c/ol\u003e\u003cp\u003eIt is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door 1, then the host's choice of which goat-hiding door to open is represented by a conditional probability matrix Ph\u003c/p\u003e\u003cpre\u003e   Ph = [ p_11   p_12   p_13\r\n          p_21   p_22   p_23\r\n          p_31   p_32   p_33 ]\u003c/pre\u003e\u003cp\u003eIn the above matrix, p_ij represents the probability that the host opens door j\r\ngiven that the car is behind door i.\u003c/p\u003e\u003cp\u003eInterpreting the matrix in terms of the standard assumptions implies\u003c/p\u003e\u003cpre\u003e p_i1 = 0            i.e.  the host cannot open door 1, the player's initial choice.  \u003c/pre\u003e\u003cpre\u003e p_i2 + pi3 = 1      i.e. the host must always open a door, 2 or 3, not initially picked by the player.\u003c/pre\u003e\u003cpre\u003e p_ii = 0            i.e. the host must always open a door to reveal a goat and never the car.\u003c/pre\u003e\u003cp\u003eOn the game show, you have initially chosen door 1 and the host, Monty Hall, opened door H (2 or 3), using the conditional probability Ph.\u003c/p\u003e\u003cp\u003eWhat is the probability \u003cb\u003ePws\u003c/b\u003e that you will win the car by switching your choice to the door remaining?\u003c/p\u003e","function_template":"function Psw = MontyHall(H, Ph)\r\n  \r\nend","test_suite":"%%\r\nH = 2;\r\nPh = [0 0.55 0.45;0 0 1;0 1 0];\r\nPws_correct = 0.6452;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.55 0.45;0 0 1;0 1 0];\r\nPws_correct = 0.6897;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.25 0.75;0 0 1;0 1 0];\r\nPws_correct = 0.8;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.25 0.75;0 0 1;0 1 0];\r\nPws_correct = 0.5714;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.5 0.5;0 0 1;0 1 0];\r\nPws_correct = 0.6667;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.5 0.5;0 0 1;0 1 0];\r\nPws_correct = 0.6667;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.80 0.20;0 0 1;0 1 0];\r\nPws_correct = 0.5556;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.80 0.20;0 0 1;0 1 0];\r\nPws_correct = 0.8333;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.60 0.40;0 0 1;0 1 0];\r\nPws_correct = 0.6250;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.60 0.40;0 0 1;0 1 0];\r\nPws_correct = 0.7143;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.23 0.77;0 0 1;0 1 0];\r\nPws_correct = 0.8130;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.23 0.77;0 0 1;0 1 0];\r\nPws_correct = 0.5650;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.27 0.73;0 0 1;0 1 0];\r\nPws_correct = 0.7874;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.27 0.73;0 0 1;0 1 0];\r\nPws_correct = 0.5780;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.90 0.10;0 0 1;0 1 0];\r\nPws_correct = 0.5263;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.90 0.10;0 0 1;0 1 0];\r\nPws_correct = 0.9091;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 1 0;0 0 1;0 1 0];\r\nPws_correct = 0.5;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 1 0;0 0 1;0 1 0];\r\nPws_correct = 1;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.37 0.63;0 0 1;0 1 0];\r\nPws_correct = 0.7299;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.37 0.63;0 0 1;0 1 0];\r\nPws_correct = 0.6135;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.01 0.99;0 0 1;0 1 0];\r\nPws_correct = 0.9901;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.01 0.99;0 0 1;0 1 0];\r\nPws_correct = 0.5025;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":178544,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2018-09-18T18:38:37.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-08-08T07:42:36.000Z","updated_at":"2018-09-18T18:38:37.000Z","published_at":"2018-08-08T08:37:59.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSuppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, \\\"Do you want to pick door No. 2?\\\" Is it to your advantage to switch your choice?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWe will now play the game using the standard assumptions:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host must always open a door that was not picked by the contestant.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host must always open a door to reveal a goat and never the car.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host must always offer the chance to switch between the originally chosen door and the remaining closed door.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door 1, then the host's choice of which goat-hiding door to open is represented by a conditional probability matrix Ph\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   Ph = [ p_11   p_12   p_13\\n          p_21   p_22   p_23\\n          p_31   p_32   p_33 ]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the above matrix, p_ij represents the probability that the host opens door j given that the car is behind door i.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInterpreting the matrix in terms of the standard assumptions implies\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ p_i1 = 0            i.e.  the host cannot open door 1, the player's initial choice.  \\n\\n p_i2 + pi3 = 1      i.e. the host must always open a door, 2 or 3, not initially picked by the player.\\n\\n p_ii = 0            i.e. the host must always open a door to reveal a goat and never the car.]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the game show, you have initially chosen door 1 and the host, Monty Hall, opened door H (2 or 3), using the conditional probability Ph.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat is the probability\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ePws\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e that you will win the car by switching your choice to the door remaining?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":44724,"title":"Let's Make A Deal: The Player's Dilemma 2","description":"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, either immediately opens your door or opens another door, say No. 3, which might have the car or a goat. If the host opens door No. 3 and it has a goat behind it, he then offers you the choice, \"Do you want to pick door No. 2?\" Is it to your advantage to switch your choice?\r\n\r\nWe will now play the game using the assumptions:\r\n\r\n# The host can open any of the doors including that picked by the player.\r\n# The host can open a door to reveal either a goat or the car.\r\n# The host doesn't always offer the chance to switch between the originally chosen \r\n  door and the closed doors.\r\n\r\nIt is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door D, then the host's choice of which door to open is represented by a conditional probability matrix Ph\r\n\r\n   Ph = [ p_11   p_12   p_13\r\n          p_21   p_22   p_23\r\n          p_31   p_32   p_33 ]\r\n\r\nIn the above matrix, p_ij represents the probability that the host opens door j given that the car is behind door i.\r\n\r\nInterpreting the matrix in terms of the assumptions implies that all elements p_ij of the conditional probability matrix can be nonzero with the constraint that\r\n\r\n  p_i1 + p_i2 + p_i3 = 1    i.e. the host must always open a door to reveal the car or a goat. \r\n\r\nOn the game show, you have initially chosen door D and the host, Monty Hall, opened door H, using the conditional probability Ph.\r\n\r\nWhat is the probability *Pws* that you will win the car by switching your choice to the door remaining?\r\n\r\n*If switching option is not available (i.e. host opens a door with the car behind it) return NaN.*\r\n","description_html":"\u003cp\u003eSuppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, either immediately opens your door or opens another door, say No. 3, which might have the car or a goat. If the host opens door No. 3 and it has a goat behind it, he then offers you the choice, \"Do you want to pick door No. 2?\" Is it to your advantage to switch your choice?\u003c/p\u003e\u003cp\u003eWe will now play the game using the assumptions:\u003c/p\u003e\u003col\u003e\u003cli\u003eThe host can open any of the doors including that picked by the player.\u003c/li\u003e\u003cli\u003eThe host can open a door to reveal either a goat or the car.\u003c/li\u003e\u003cli\u003eThe host doesn't always offer the chance to switch between the originally chosen \r\n  door and the closed doors.\u003c/li\u003e\u003c/ol\u003e\u003cp\u003eIt is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door D, then the host's choice of which door to open is represented by a conditional probability matrix Ph\u003c/p\u003e\u003cpre\u003e   Ph = [ p_11   p_12   p_13\r\n          p_21   p_22   p_23\r\n          p_31   p_32   p_33 ]\u003c/pre\u003e\u003cp\u003eIn the above matrix, p_ij represents the probability that the host opens door j given that the car is behind door i.\u003c/p\u003e\u003cp\u003eInterpreting the matrix in terms of the assumptions implies that all elements p_ij of the conditional probability matrix can be nonzero with the constraint that\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003ep_i1 + p_i2 + p_i3 = 1    i.e. the host must always open a door to reveal the car or a goat. \r\n\u003c/pre\u003e\u003cp\u003eOn the game show, you have initially chosen door D and the host, Monty Hall, opened door H, using the conditional probability Ph.\u003c/p\u003e\u003cp\u003eWhat is the probability \u003cb\u003ePws\u003c/b\u003e that you will win the car by switching your choice to the door remaining?\u003c/p\u003e\u003cp\u003e\u003cb\u003eIf switching option is not available (i.e. host opens a door with the car behind it) return NaN.\u003c/b\u003e\u003c/p\u003e","function_template":"function Pws = MontyHall2(D,H,Ph)\r\n  Pws = ;\r\nend","test_suite":"\r\n%% CASE 1: Host always opens the door containing the car.\r\n%      Ph = [1   0   0\r\n%            0   1   0\r\n%            0   0   1]\r\n% In this case, the host does not offer the player the choice of switching\r\n% doors. Hence, return Pws = NaN.\r\n%__________________________________________________________________________\r\nD = 1;\r\nH = 2;\r\nPh = diag([1 1 1]);\r\nPws_correct = NaN;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 1;\r\nH = 3;\r\nPh = diag([1 1 1]);\r\nPws_correct = NaN;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = diag([1 1 1]);\r\nPws_correct = NaN;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n\r\n%% CASE 2: Host (uniformly) randomly opens any door.\r\n%      Ph = [1/3  1/3  1/3\r\n%            1/3  1/3  1/3\r\n%            1/3  1/3  1/3]\r\n% In this case, the game is fair because the host does not make any effort\r\n% to open any particular door. Hence, the chance of winning or losing by\r\n% switching is equal i.e. Psw = 0.5.\r\n%__________________________________________________________________________\r\n\r\nD = 1;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPws_correct = 0.5;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 1;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPws_correct = 0.5;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 1;\r\nPh = ones(3)/3;\r\nPws_correct = 0.5;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPws_correct = 0.5;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%% CASE 3: Host opens door 2 if the car is not behind it. Otherwise,\r\n% the host opens door 3.\r\n%      Ph = [0   1   0\r\n%            0   0   1\r\n%            0   1   0]\r\n%__________________________________________________________________________\r\n\r\nD = 1;\r\nH = 3;\r\nPh = [0 1 0;\r\n      0 0 1;\r\n      0 1 0];\r\nPws_correct = 1;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 1;\r\nH = 2;\r\nPh = [0 1 0;\r\n      0 0 1;\r\n      0 1 0];\r\nPws_correct = 0.5;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0 1 0;\r\n      0 0 1;\r\n      0 1 0];\r\nPws_correct = 0;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [0 1 0;\r\n      0 0 1;\r\n      0 1 0];\r\nPws_correct = 0.5;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%% CASE 4: Host opens the door based on the standard assumptions of \r\n%  Marilyn vos Savant.\r\n%__________________________________________________________________________\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPws_correct = 0.6667;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPws_correct = 0.6667;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPws_correct = 0.3333;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%% CASE 5: Host opens the door based on a completely random approach.\r\n%__________________________________________________________________________\r\n\r\nD = 1;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPws_correct = 0.5045;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPws_correct = 0.4955;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPws_correct = 0.4745;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [0.5000    0.1000    0.4000\r\n      0.0500    0.5000    0.4500\r\n      0.3333    0.0667    0.6000];\r\nPws_correct = 0.8696;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0.5000    0.1000    0.4000\r\n      0.0500    0.5000    0.4500\r\n      0.3333    0.0667    0.6000];\r\nPws_correct = 0.4706;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [0.5000    0.1000    0.4000\r\n      0.0500    0.5000    0.4500\r\n      0.3333    0.0667    0.6000];\r\nPws_correct = 0.1304;\r\nPws = MontyHall2(D,H,Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":178544,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2018-08-10T08:50:31.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2018-08-08T10:20:24.000Z","updated_at":"2018-08-10T08:50:31.000Z","published_at":"2018-08-08T12:55:34.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSuppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, either immediately opens your door or opens another door, say No. 3, which might have the car or a goat. If the host opens door No. 3 and it has a goat behind it, he then offers you the choice, \\\"Do you want to pick door No. 2?\\\" Is it to your advantage to switch your choice?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWe will now play the game using the assumptions:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host can open any of the doors including that picked by the player.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host can open a door to reveal either a goat or the car.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host doesn't always offer the chance to switch between the originally chosen door and the closed doors.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door D, then the host's choice of which door to open is represented by a conditional probability matrix Ph\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   Ph = [ p_11   p_12   p_13\\n          p_21   p_22   p_23\\n          p_31   p_32   p_33 ]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the above matrix, p_ij represents the probability that the host opens door j given that the car is behind door i.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInterpreting the matrix in terms of the assumptions implies that all elements p_ij of the conditional probability matrix can be nonzero with the constraint that\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[p_i1 + p_i2 + p_i3 = 1    i.e. the host must always open a door to reveal the car or a goat.]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the game show, you have initially chosen door D and the host, Monty Hall, opened door H, using the conditional probability Ph.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat is the probability\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ePws\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e that you will win the car by switching your choice to the door remaining?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eIf switching option is not available (i.e. host opens a door with the car behind it) return NaN.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44727,"title":"Let's Make A Deal: Criminal Minds","description":"The game of Let's Make A Deal proceeds as follows;\r\n\r\nThe player is given the choice of three doors: Behind one door is a car; behind the others, goats. The player picks a door, say No. 1, and the host, either immediately opens that door or opens another door, say No. 3, which might have the car or a goat. If the host opens door No. 3 and it has a goat behind it, he then offers the player the choice, \"Do you want to pick door No. 2?\"\r\n\r\nThe game play is guided by the following set of rules:\r\n\r\n# The host can open any of the doors including that picked by the player.\r\n# The host can open a door to reveal either a goat or the car.\r\n# The host doesn't always offer the chance to switch between the originally chosen door and the closed doors.\r\n\r\nIn order to manipulate the game play, the host does not always (uniformly) randomly hide the car behind the doors. Instead, based on previous games statistics, the host determines the probability Pc. \r\n\r\n Pc = [pc_1  pc_2  pc_3]   pc_i is the probability of the host placing car behind door i. \r\n\r\nIf the player initially picks door D, then the host's choice of which door to open is represented by a conditional probability matrix Ph\r\n\r\n   Ph = [ ph_11   ph_12   ph_13\r\n          ph_21   ph_22   ph_23\r\n          ph_31   ph_32   ph_33 ]\r\n\r\nIn the above matrix, ph_ij represents the probability that the host opens door j given that the car is behind door i.\r\n\r\nInterpreting the matrix in terms of the assumptions implies that all elements ph_ij of the conditional probability matrix can be nonzero with the constraint that\r\n\r\n ph_i1 + ph_i2 + ph_i3 = 1    i.e. the host must always open a door to reveal the car or a goat. \r\n\r\nOn the game show, the host places the car behind a door with probability Pc in order to manipulate the outcome. The player then choose door D and the host, Monty Hall, opens door H using the conditional probability Ph.\r\n\r\nIf the door H opened by the host has a goat behind it, what is the probability *Pws* that the player will win the car by switching the initial door choice to the door remaining?\r\n\r\n*If switching option is not available (i.e. host always opens a door with the car behind it) return NaN.*\r\n","description_html":"\u003cp\u003eThe game of Let's Make A Deal proceeds as follows;\u003c/p\u003e\u003cp\u003eThe player is given the choice of three doors: Behind one door is a car; behind the others, goats. The player picks a door, say No. 1, and the host, either immediately opens that door or opens another door, say No. 3, which might have the car or a goat. If the host opens door No. 3 and it has a goat behind it, he then offers the player the choice, \"Do you want to pick door No. 2?\"\u003c/p\u003e\u003cp\u003eThe game play is guided by the following set of rules:\u003c/p\u003e\u003col\u003e\u003cli\u003eThe host can open any of the doors including that picked by the player.\u003c/li\u003e\u003cli\u003eThe host can open a door to reveal either a goat or the car.\u003c/li\u003e\u003cli\u003eThe host doesn't always offer the chance to switch between the originally chosen door and the closed doors.\u003c/li\u003e\u003c/ol\u003e\u003cp\u003eIn order to manipulate the game play, the host does not always (uniformly) randomly hide the car behind the doors. Instead, based on previous games statistics, the host determines the probability Pc.\u003c/p\u003e\u003cpre\u003e Pc = [pc_1  pc_2  pc_3]   pc_i is the probability of the host placing car behind door i. \u003c/pre\u003e\u003cp\u003eIf the player initially picks door D, then the host's choice of which door to open is represented by a conditional probability matrix Ph\u003c/p\u003e\u003cpre\u003e   Ph = [ ph_11   ph_12   ph_13\r\n          ph_21   ph_22   ph_23\r\n          ph_31   ph_32   ph_33 ]\u003c/pre\u003e\u003cp\u003eIn the above matrix, ph_ij represents the probability that the host opens door j given that the car is behind door i.\u003c/p\u003e\u003cp\u003eInterpreting the matrix in terms of the assumptions implies that all elements ph_ij of the conditional probability matrix can be nonzero with the constraint that\u003c/p\u003e\u003cpre\u003e ph_i1 + ph_i2 + ph_i3 = 1    i.e. the host must always open a door to reveal the car or a goat. \u003c/pre\u003e\u003cp\u003eOn the game show, the host places the car behind a door with probability Pc in order to manipulate the outcome. The player then choose door D and the host, Monty Hall, opens door H using the conditional probability Ph.\u003c/p\u003e\u003cp\u003eIf the door H opened by the host has a goat behind it, what is the probability \u003cb\u003ePws\u003c/b\u003e that the player will win the car by switching the initial door choice to the door remaining?\u003c/p\u003e\u003cp\u003e\u003cb\u003eIf switching option is not available (i.e. host always opens a door with the car behind it) return NaN.\u003c/b\u003e\u003c/p\u003e","function_template":"function Pws = MontyHall3(D,H,Ph,Pc)\r\n  \r\nend","test_suite":"%% CASE 1: Monty Hall (uniformly) randomly opens any door.\r\n%      Ph = [1/3  1/3  1/3\r\n%            1/3  1/3  1/3\r\n%            1/3  1/3  1/3]\r\n%__________________________________________________________________________\r\n%\r\n% CASE 1A: Host always hides the car behind one particular door.\r\n%   Pc = [1  0  0] or [0  1  0]  or  [0  0  1]\r\n%__________________________________________________________________________\r\n% 1. Pc = [1 0 0], D = 1,  H = 3. \r\n% The host always hides the car behind door 1.\r\n% Consequently, a player with an initial door choice of D = 1 cannot win\r\n% by switching doors after the host opens door H = 3 i.e. Psw = 0.\r\nD = 1;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPc = [1  0  0];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n% 2. Pc = [1 0 0], D = 1, H = 2.\r\n% The host always hides the car behind door 1.\r\n% Consequently, a player with an initial door choice of D = 1 cannot win\r\n% by switching doors after the host opens door H = 2 i.e. Psw = 0.\r\nD = 1;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [1  0  0];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n% 3. Pc = [0 1 0], D = 1, H = 2. \r\n% The host always hides the car behind door 2.\r\n% Consequently, a player with an initial door choice of D = 1 does not\r\n% have the option to switch doors after the host opens door H = 2.\r\nD = 1;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [0  1  0];\r\nPws_correct = NaN;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n% 4. Pc = [0 1 0], D = 1, H = 3. \r\n% The host always hides the car behind door 2.\r\n% Consequently, a player with an initial door choice of D = 1 can only win\r\n% by switching to door 2 after the host opens door H = 3. i.e. Psw = 1.\r\nD = 1;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPc = [0  1  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n% 5. Pc = [0 0 1], D = 1, H = 2\r\n% The host always hides the car behind door 3.\r\n% Consequently, a player with an initial door choice of D = 1 does not\r\n% have the option to switch doors after the host opens door H = 2.\r\nD = 1;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [0  0  1];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n\r\n% 6. Pc = [0 0 1], D = 1, H = 3\r\n% The host always hides the car behind door 3.\r\n% Consequently, a player with an initial door choice of D = 1 does not\r\n% have the option to switch doors after the host opens door H = 3.\r\nD = 1;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPc = [0  0  1];\r\nPws_correct = NaN;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 1B: Host hides the car behind two particular doors with equal probability.\r\n% Pc = [0  0.5  0.5]  or [0.5  0.5  0]  or  [0.5  0  0.5]\r\n%__________________________________________________________________________\r\n% 1. Pc = [0  0.5  0.5], D = 3, H = 1.\r\n% The car is either behind door 2 or 3. If the host opens door H = 1, the\r\n% player's chance of winning by switching from door D = 3 to door 2 is\r\n% equal i.e. Psw = 0.5.\r\nD = 3;\r\nH = 1;\r\nPh = ones(3)/3;\r\nPc = [0  0.5  0.5];\r\nPws_correct = 0.5;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n% 2. Pc = [0  0.5  0.5], D = 3, H = 2.\r\n% The host never hides the car behind door 1. Hence, the player can never\r\n% wi by switching to door 1.\r\nD = 3;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [0  0.5  0.5];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n% 3. Pc = [0.5  0.5  0], D = 3, H = 1.\r\n% The host never hides the car behind door 3. Hence, the player can only \r\n% win by switching to door 2.\r\nD = 3;\r\nH = 1;\r\nPh = ones(3)/3;\r\nPc = [0.5  0.5  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n% 4. Pc = [0.5  0.5  0], D = 3, H = 2.\r\n% The host never hides the car behind door 3. Hence, the player can only \r\n% win by switching to door 1.\r\nD = 3;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [0.5  0.5  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n% 5. Pc = [0.5  0  0.5], D = 3, H = 1.  \r\n% The host never hides the car behind door 2. Hence, the player can\r\n% never win by switching to door 2.\r\nD = 3;\r\nH = 1;\r\nPh = ones(3)/3;\r\nPc = [0.5  0  0.5];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 1C: Hosts hides the car behind any of the three doors with equal probability.\r\n%   Pc = [1/3  1/3  1/3]\r\n%__________________________________________________________________________\r\n% The host does not manipulate the game. Hence, it is a fair game.\r\nD = 1;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPc = [1  1  1]/3;\r\nPws_correct = 0.5;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 1D: Host hides the car with a pre-determined probability.\r\n%   Pc = [0.26  0.43  0.31]\r\n%__________________________________________________________________________\r\n\r\nD = 1;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.6232;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 1;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.5439;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 1;\r\nPh = ones(3)/3;\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.4189;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = ones(3)/3;\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.3768;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = ones(3)/3;\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.5811;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = ones(3)/3;\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.4561;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%% CASE 2: Monty Hall opens the door based on the standard assumptions of \r\n%  Marilyn vos Savant.\r\n%      Ph = [ 0    1   0\r\n%             1    0   0\r\n%            0.5  0.5  0]\r\n%__________________________________________________________________________\r\n%\r\n% CASE 2A: Host always hides the car behind one particular door.\r\n%   Pc = [1  0  0] or [0  1  0]  or  [0  0  1]\r\n%__________________________________________________________________________\r\n \r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [1  0  0];\r\nPws_correct = NaN;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [1  0  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0  1  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0  1  0];\r\nPws_correct = NaN;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0  0  1];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0  0  1];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 2B: Host hides the car behind two particular doors with equal \r\n% probability.\r\n% Pc = [0  0.5  0.5]  or [0.5  0.5  0]  or  [0.5  0  0.5]\r\n%__________________________________________________________________________\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.5  0.5  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.5  0.5  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0  0.5  0.5];\r\nPws_correct = 0.6667;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0  0.5  0.5];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.5  0  0.5];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.5  0  0.5];\r\nPws_correct = 0.6667;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 2C: Hosts hides the car behind any of the three doors with equal \r\n% probability.\r\n%   Pc = [1/3  1/3  1/3]\r\n%__________________________________________________________________________\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [1  1  1]/3;\r\nPws_correct = 0.6667;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [1  1  1]/3;\r\nPws_correct = 0.6667;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 2D: Host hides the car with a pre-determined probability.\r\n%   Pc = [0.26  0.43  0.31]\r\n%__________________________________________________________________________\r\n\r\nD = 1;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.3735;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.2650;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.7350;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [  0    1   0;\r\n        1    0   0;\r\n       0.5  0.5  0];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.6265;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n\r\n%% CASE 3: Host opens the door based on a completely randomized approach.\r\n%__________________________________________________________________________\r\n%\r\n% CASE 3A: Host always hides the car behind one particular door.\r\n%   Pc = [1  0  0] or [0  1  0]  or  [0  0  1]\r\n%__________________________________________________________________________\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [1  0  0];\r\nPws_correct = NaN;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [1  0  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0  1  0];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0  0  1];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0  0  1];\r\nPws_correct = NaN;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 3B: Host hides the car behind two particular doors with equal probability.\r\n% Pc = [0  0.5  0.5]  or [0.5  0.5  0]  or  [0.5  0  0.5]\r\n%__________________________________________________________________________\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0  0.5  0.5];\r\nPws_correct = 0.6136;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0  0.5  0.5];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.5  0.5  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.5  0.5  0];\r\nPws_correct = 1;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.5  0  0.5];\r\nPws_correct = 0;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 3C: Hosts hides the car behind any of the three doors with equal \r\n% probability.\r\n%   Pc = [1/3  1/3  1/3]\r\n%__________________________________________________________________________\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [1  1  1]/3;\r\nPws_correct = 0.6136;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [1  1  1]/3;\r\nPws_correct = 0.4745;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n%__________________________________________________________________________\r\n%\r\n% CASE 3D: Host hides the car with a pre-determined probability.\r\n%   Pc = [0.26  0.43  0.31]\r\n%__________________________________________________________________________\r\n\r\nD = 1;\r\nH = 2;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.5690;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 1;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.6274;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.3123;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 2;\r\nH = 3;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.3726;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 1;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.6877;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\nD = 3;\r\nH = 2;\r\nPh = [0.3333    0.3651   0.3016\r\n      0.4646    0.2283   0.3071\r\n      0.2926    0.4043   0.3032];\r\nPc = [0.26  0.43  0.31];\r\nPws_correct = 0.4310;\r\nPws = MontyHall3(D,H,Ph,Pc);\r\nassert(isequaln(round(Pws,4),Pws_correct))\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":1,"created_by":178544,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2018-08-10T13:05:15.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-08-09T10:04:50.000Z","updated_at":"2018-08-10T13:05:15.000Z","published_at":"2018-08-10T13:00:57.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe game of Let's Make A Deal proceeds as follows;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe player is given the choice of three doors: Behind one door is a car; behind the others, goats. The player picks a door, say No. 1, and the host, either immediately opens that door or opens another door, say No. 3, which might have the car or a goat. If the host opens door No. 3 and it has a goat behind it, he then offers the player the choice, \\\"Do you want to pick door No. 2?\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe game play is guided by the following set of rules:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host can open any of the doors including that picked by the player.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host can open a door to reveal either a goat or the car.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host doesn't always offer the chance to switch between the originally chosen door and the closed doors.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn order to manipulate the game play, the host does not always (uniformly) randomly hide the car behind the doors. Instead, based on previous games statistics, the host determines the probability Pc.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Pc = [pc_1  pc_2  pc_3]   pc_i is the probability of the host placing car behind door i.]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf the player initially picks door D, then the host's choice of which door to open is represented by a conditional probability matrix Ph\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   Ph = [ ph_11   ph_12   ph_13\\n          ph_21   ph_22   ph_23\\n          ph_31   ph_32   ph_33 ]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the above matrix, ph_ij represents the probability that the host opens door j given that the car is behind door i.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInterpreting the matrix in terms of the assumptions implies that all elements ph_ij of the conditional probability matrix can be nonzero with the constraint that\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ ph_i1 + ph_i2 + ph_i3 = 1    i.e. the host must always open a door to reveal the car or a goat.]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the game show, the host places the car behind a door with probability Pc in order to manipulate the outcome. The player then choose door D and the host, Monty Hall, opens door H using the conditional probability Ph.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf the door H opened by the host has a goat behind it, what is the probability\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ePws\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e that the player will win the car by switching the initial door choice to the door remaining?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eIf switching option is not available (i.e. host always opens a door with the car behind it) return NaN.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44723,"title":"Let's Make A Deal: The Player's Dilemma ","description":"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, \"Do you want to pick door No. 2?\" Is it to your advantage to switch your choice?\r\n\r\nWe will now play the game using the standard assumptions:\r\n\r\n# The host must always open a door that was not picked by the contestant.\r\n# The host must always open a door to reveal a goat and never the car.\r\n# The host must always offer the chance to switch between the originally chosen door and the remaining closed door.\r\n\r\nIt is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door 1, then the host's choice of which goat-hiding door to open is represented by a conditional probability matrix Ph \r\n     \r\n   Ph = [ p_11   p_12   p_13\r\n          p_21   p_22   p_23\r\n          p_31   p_32   p_33 ]\r\n\r\nIn the above matrix, p_ij represents the probability that the host opens door j\r\ngiven that the car is behind door i. \r\n\r\nInterpreting the matrix in terms of the standard assumptions implies \r\n\r\n p_i1 = 0            i.e.  the host cannot open door 1, the player's initial choice.  \r\n  \r\n p_i2 + pi3 = 1      i.e. the host must always open a door, 2 or 3, not initially picked by the player.\r\n\r\n p_ii = 0            i.e. the host must always open a door to reveal a goat and never the car.\r\n\r\nOn the game show, you have initially chosen door 1 and the host, Monty Hall, opened door H (2 or 3), using the conditional probability Ph. \r\n\r\nWhat is the probability *Pws* that you will win the car by switching your choice to the door remaining? ","description_html":"\u003cp\u003eSuppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, \"Do you want to pick door No. 2?\" Is it to your advantage to switch your choice?\u003c/p\u003e\u003cp\u003eWe will now play the game using the standard assumptions:\u003c/p\u003e\u003col\u003e\u003cli\u003eThe host must always open a door that was not picked by the contestant.\u003c/li\u003e\u003cli\u003eThe host must always open a door to reveal a goat and never the car.\u003c/li\u003e\u003cli\u003eThe host must always offer the chance to switch between the originally chosen door and the remaining closed door.\u003c/li\u003e\u003c/ol\u003e\u003cp\u003eIt is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door 1, then the host's choice of which goat-hiding door to open is represented by a conditional probability matrix Ph\u003c/p\u003e\u003cpre\u003e   Ph = [ p_11   p_12   p_13\r\n          p_21   p_22   p_23\r\n          p_31   p_32   p_33 ]\u003c/pre\u003e\u003cp\u003eIn the above matrix, p_ij represents the probability that the host opens door j\r\ngiven that the car is behind door i.\u003c/p\u003e\u003cp\u003eInterpreting the matrix in terms of the standard assumptions implies\u003c/p\u003e\u003cpre\u003e p_i1 = 0            i.e.  the host cannot open door 1, the player's initial choice.  \u003c/pre\u003e\u003cpre\u003e p_i2 + pi3 = 1      i.e. the host must always open a door, 2 or 3, not initially picked by the player.\u003c/pre\u003e\u003cpre\u003e p_ii = 0            i.e. the host must always open a door to reveal a goat and never the car.\u003c/pre\u003e\u003cp\u003eOn the game show, you have initially chosen door 1 and the host, Monty Hall, opened door H (2 or 3), using the conditional probability Ph.\u003c/p\u003e\u003cp\u003eWhat is the probability \u003cb\u003ePws\u003c/b\u003e that you will win the car by switching your choice to the door remaining?\u003c/p\u003e","function_template":"function Psw = MontyHall(H, Ph)\r\n  \r\nend","test_suite":"%%\r\nH = 2;\r\nPh = [0 0.55 0.45;0 0 1;0 1 0];\r\nPws_correct = 0.6452;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.55 0.45;0 0 1;0 1 0];\r\nPws_correct = 0.6897;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.25 0.75;0 0 1;0 1 0];\r\nPws_correct = 0.8;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.25 0.75;0 0 1;0 1 0];\r\nPws_correct = 0.5714;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.5 0.5;0 0 1;0 1 0];\r\nPws_correct = 0.6667;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.5 0.5;0 0 1;0 1 0];\r\nPws_correct = 0.6667;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.80 0.20;0 0 1;0 1 0];\r\nPws_correct = 0.5556;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.80 0.20;0 0 1;0 1 0];\r\nPws_correct = 0.8333;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.60 0.40;0 0 1;0 1 0];\r\nPws_correct = 0.6250;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.60 0.40;0 0 1;0 1 0];\r\nPws_correct = 0.7143;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.23 0.77;0 0 1;0 1 0];\r\nPws_correct = 0.8130;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.23 0.77;0 0 1;0 1 0];\r\nPws_correct = 0.5650;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.27 0.73;0 0 1;0 1 0];\r\nPws_correct = 0.7874;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.27 0.73;0 0 1;0 1 0];\r\nPws_correct = 0.5780;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.90 0.10;0 0 1;0 1 0];\r\nPws_correct = 0.5263;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.90 0.10;0 0 1;0 1 0];\r\nPws_correct = 0.9091;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 1 0;0 0 1;0 1 0];\r\nPws_correct = 0.5;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 1 0;0 0 1;0 1 0];\r\nPws_correct = 1;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.37 0.63;0 0 1;0 1 0];\r\nPws_correct = 0.7299;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.37 0.63;0 0 1;0 1 0];\r\nPws_correct = 0.6135;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 2;\r\nPh = [0 0.01 0.99;0 0 1;0 1 0];\r\nPws_correct = 0.9901;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n\r\n%%\r\nH = 3;\r\nPh = [0 0.01 0.99;0 0 1;0 1 0];\r\nPws_correct = 0.5025;\r\nPws = MontyHall(H, Ph);\r\nassert(isequal(round(Pws,4),Pws_correct))\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":178544,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2018-09-18T18:38:37.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-08-08T07:42:36.000Z","updated_at":"2018-09-18T18:38:37.000Z","published_at":"2018-08-08T08:37:59.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSuppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, \\\"Do you want to pick door No. 2?\\\" Is it to your advantage to switch your choice?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWe will now play the game using the standard assumptions:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host must always open a door that was not picked by the contestant.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host must always open a door to reveal a goat and never the car.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe host must always offer the chance to switch between the originally chosen door and the remaining closed door.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt is also typically presumed that the car is initially hidden randomly behind the doors and that, if the player initially picks door 1, then the host's choice of which goat-hiding door to open is represented by a conditional probability matrix Ph\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   Ph = [ p_11   p_12   p_13\\n          p_21   p_22   p_23\\n          p_31   p_32   p_33 ]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the above matrix, p_ij represents the probability that the host opens door j given that the car is behind door i.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInterpreting the matrix in terms of the standard assumptions implies\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ p_i1 = 0            i.e.  the host cannot open door 1, the player's initial choice.  \\n\\n p_i2 + pi3 = 1      i.e. the host must always open a door, 2 or 3, not initially picked by the player.\\n\\n p_ii = 0            i.e. the host must always open a door to reveal a goat and never the car.]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the game show, you have initially chosen door 1 and the host, Monty Hall, opened door H (2 or 3), using the conditional probability Ph.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat is the probability\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ePws\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e that you will win the car by switching your choice to the door remaining?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"term":"tag:\"game show\"","current_player_id":null,"fields":[{"name":"page","type":"integer","callback":null,"default":1,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"per_page","type":"integer","callback":null,"default":50,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"sort","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"body","type":"text","callback":null,"default":"*:*","directive":null,"facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":false},{"name":"group","type":"string","callback":null,"default":null,"directive":"group","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"difficulty_rating_bin","type":"string","callback":null,"default":null,"directive":"difficulty_rating_bin","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"id","type":"integer","callback":null,"default":null,"directive":"id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"tag","type":"string","callback":null,"default":null,"directive":"tag","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"product","type":"string","callback":null,"default":null,"directive":"product","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_at","type":"timeframe","callback":{},"default":null,"directive":"created_at","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"profile_id","type":"integer","callback":null,"default":null,"directive":"author_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_by","type":"string","callback":null,"default":null,"directive":"author","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player_id","type":"integer","callback":null,"default":null,"directive":"solver_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player","type":"string","callback":null,"default":null,"directive":"solver","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"solvers_count","type":"integer","callback":null,"default":null,"directive":"solvers_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"comments_count","type":"integer","callback":null,"default":null,"directive":"comments_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"likes_count","type":"integer","callback":null,"default":null,"directive":"likes_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leader_id","type":"integer","callback":null,"default":null,"directive":"leader_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leading_solution","type":"integer","callback":null,"default":null,"directive":"leading_solution","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true}],"filters":[{"name":"asset_type","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":"\"cody:problem\"","prepend":true},{"name":"profile_id","type":"integer","callback":{},"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":"author_id","static":null,"prepend":true}],"query":{"params":{"per_page":50,"term":"tag:\"game show\"","current_player":null,"sort":"map(difficulty_value,0,0,999) asc"},"parser":"MathWorks::Search::Solr::QueryParser","directives":{"term":{"directives":{"tag":[["tag:\"game show\"","","\"","game show","\""]]}}},"facets":{"#\u003cMathWorks::Search::Field:0x00007f1940af5f18\u003e":null,"#\u003cMathWorks::Search::Field:0x00007f1940af5e78\u003e":null},"filters":{"#\u003cMathWorks::Search::Field:0x00007f1940af55b8\u003e":"\"cody:problem\""},"fields":{"#\u003cMathWorks::Search::Field:0x00007f1940af6198\u003e":1,"#\u003cMathWorks::Search::Field:0x00007f1940af60f8\u003e":50,"#\u003cMathWorks::Search::Field:0x00007f1940af6058\u003e":"map(difficulty_value,0,0,999) asc","#\u003cMathWorks::Search::Field:0x00007f1940af5fb8\u003e":"tag:\"game show\""},"user_query":{"#\u003cMathWorks::Search::Field:0x00007f1940af5fb8\u003e":"tag:\"game show\""},"queried_facets":{}},"query_backend":{"connection":{"configuration":{"index_url":"http://index-op-v2/solr/","query_url":"http://search-op-v2/solr/","direct_access_index_urls":["http://index-op-v2/solr/"],"direct_access_query_urls":["http://search-op-v2/solr/"],"timeout":10,"vhost":"search","exchange":"search.topic","heartbeat":30,"pre_index_mode":false,"host":"rabbitmq-eks","port":5672,"username":"search","password":"J3bGPZzQ7asjJcCk","virtual_host":"search","indexer":"amqp","http_logging":"true","core":"cody"},"query_connection":{"uri":"http://search-op-v2/solr/cody/","proxy":null,"connection":{"parallel_manager":null,"headers":{"User-Agent":"Faraday v1.0.1"},"params":{},"options":{"params_encoder":"Faraday::FlatParamsEncoder","proxy":null,"bind":null,"timeout":null,"open_timeout":null,"read_timeout":null,"write_timeout":null,"boundary":null,"oauth":null,"context":null,"on_data":null},"ssl":{"verify":true,"ca_file":null,"ca_path":null,"verify_mode":null,"cert_store":null,"client_cert":null,"client_key":null,"certificate":null,"private_key":null,"verify_depth":null,"version":null,"min_version":null,"max_version":null},"default_parallel_manager":null,"builder":{"adapter":{"name":"Faraday::Adapter::NetHttp","args":[],"block":null},"handlers":[{"name":"Faraday::Response::RaiseError","args":[],"block":null}],"app":{"app":{"ssl_cert_store":{"verify_callback":null,"error":null,"error_string":null,"chain":null,"time":null},"app":{},"connection_options":{},"config_block":null}}},"url_prefix":"http://search-op-v2/solr/cody/","manual_proxy":false,"proxy":null},"update_format":"RSolr::JSON::Generator","update_path":"update","options":{"url":"http://search-op-v2/solr/cody"}}},"query":{"params":{"per_page":50,"term":"tag:\"game show\"","current_player":null,"sort":"map(difficulty_value,0,0,999) asc"},"parser":"MathWorks::Search::Solr::QueryParser","directives":{"term":{"directives":{"tag":[["tag:\"game show\"","","\"","game show","\""]]}}},"facets":{"#\u003cMathWorks::Search::Field:0x00007f1940af5f18\u003e":null,"#\u003cMathWorks::Search::Field:0x00007f1940af5e78\u003e":null},"filters":{"#\u003cMathWorks::Search::Field:0x00007f1940af55b8\u003e":"\"cody:problem\""},"fields":{"#\u003cMathWorks::Search::Field:0x00007f1940af6198\u003e":1,"#\u003cMathWorks::Search::Field:0x00007f1940af60f8\u003e":50,"#\u003cMathWorks::Search::Field:0x00007f1940af6058\u003e":"map(difficulty_value,0,0,999) asc","#\u003cMathWorks::Search::Field:0x00007f1940af5fb8\u003e":"tag:\"game show\""},"user_query":{"#\u003cMathWorks::Search::Field:0x00007f1940af5fb8\u003e":"tag:\"game show\""},"queried_facets":{}},"options":{"fields":["id","difficulty_rating"]},"join":" "},"results":[{"id":44724,"difficulty_rating":"unrated"},{"id":44727,"difficulty_rating":"unrated"},{"id":44723,"difficulty_rating":"unrated"}]}}