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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":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 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