Problem 44723. Let's Make A Deal: The Player's Dilemma

Created by J. S. Kowontan

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{"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":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>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?</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>We will now play the game using the standard assumptions:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr></w:pPr><w:r><w:t>The host must always open a door that was not picked by the contestant.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr></w:pPr><w:r><w:t>The host must always open a door to reveal a goat and never the car.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr></w:pPr><w:r><w:t>The host must always offer the chance to switch between the originally chosen door and the remaining closed door.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>It 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</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ Ph = [ p_11 p_12 p_13\n p_21 p_22 p_23\n p_31 p_32 p_33 ]]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>In the above matrix, p_ij represents the probability that the host opens door j given that the car is behind door i.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Interpreting the matrix in terms of the standard assumptions implies</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![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.]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>On 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.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>What is the probability</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>Pws</w:t></w:r><w:r><w:t> that you will win the car by switching your choice to the door remaining?</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

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?We will now play the game using the standard assumptions:<ol><li>The host must always open a door that was not picked by the contestant.</li><li>The host must always open a door to reveal a goat and never the car.</li><li>The host must always offer the chance to switch between the originally chosen door and the remaining closed door.</li></ol>It 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<pre> Ph = [ p_11 p_12 p_13
 p_21 p_22 p_23
 p_31 p_32 p_33 ]</pre>In the above matrix, p_ij represents the probability that the host opens door j
given that the car is behind door i.Interpreting the matrix in terms of the standard assumptions implies<pre> p_i1 = 0 i.e. the host cannot open door 1, the player's initial choice. </pre><pre> p_i2 + pi3 = 1 i.e. the host must always open a door, 2 or 3, not initially picked by the player.</pre><pre> p_ii = 0 i.e. the host must always open a door to reveal a goat and never the car.</pre>On 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.What is the probability Pws that you will win the car by switching your choice to the door remaining?

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Problem 44723. Let's Make A Deal: The Player's Dilemma

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Problem 44723. Let's Make A Deal: The Player's Dilemma

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