ps1_solutions - MS&E 221 CA Erick Delage Problem Set 1...

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MS&E 221 Problem Set 1 Solutions CA: Erick Delage January 24, 2007 Problem 1 This is a variant of the famous Monty Hall problem. Let E A , E B and E C denote the events “A is to be executed”, “B is to be executed” and “C is to be executed” respectively. Let S B denote the event “the jailer tells A that B is to be freed”. Note that E A intersectiontext E B = E B intersectiontext E C = E A intersectiontext E C = . Also, P ( S B | E B ) = 0 and P ( S B | E C ) = 1 . We also know P ( E i ) = 1 / 3 for i = A, B, C . Let α = P ( S B | E A ) . Notice that α is not specified in the statement of the problem: we haven’t been told how will the jailor behave when he can choose whether to tell A that B or C will go free. P ( E A | S B ) = P ( S B | E A ) P ( E A ) P ( S B | E A ) P ( E A ) + P ( S B | E B ) P ( E B ) + P ( S B | E C ) P ( E C ) = α · 1 / 3 α · 1 / 3 + 0 · 1 / 3 + 1 · 1 / 3 = α 1 + α If we assume that α = 1 / 2 (i.e. when both B and C are to be freed the jailer flips a fair coin to decide which name to tell A), then the jailer’s reasoning is wrong: the information provided by the jailer does not affect A’s estimate of his likelihood of being executed ( P ( E A | S B ) = P ( E A ) = 1 / 3) ). On the other hand, if α negationslash = 1 / 2 , then the information given by the jailer does affect A’s estimate of his likelihood of being executed. In particular, if α

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