Question

Example 5.26:

, B 5.26. Consider the three prisoners scenario described in Example 5.26. Let A and C denote respectively the events that Alice is to be released, Bob is to be released, and Carl is to be released, which we assume to be ec Pr(B) Pr(C)-. Also let J be the event that the jailer tells Aice that Bob is to stay in jail. (a) Compute the values of Pr(B | J), Pr(J | B), and Pr(J | C) (b) Compute the values of Pr(J | Ac) and Pr(Je | Ac), where the event Ac is the event that Alice stays in jail. (c) Suppose that if Alice is the one who is to be released, then the jailer flips a fair coin to decide whether to tell Alice that Bob stays in jail or that Carl stays in jail. What is the value of Pr(A | J)? (d) Suppose instead that if Alice is the one who is to be released, then the jailer always tells her that Bob will stay in jail. Now what is the value of Pr(A| J)? Erample 5.26 (The Three Prisoners Problem). The three prisoners problem is a classical problem about conditional probability. Three prisoners, Alice, Bob, and Carl, are informed by their jailer that the next day, one of them will be released from prison, but that the other two will have to serve life sentences. The jailer says that he will not tell any prisoner what will happen to him or her. But Alice, who reasons that her chances of going free are now j, asks the jailer to give her the name of one prisoner, other than herself, who will not go free. The jailer tells Alice that Bob will remain in jail. Now what are Alice's chances of going free? Has the probability changed? Alice could argue that she now has achance of going free, since Bob will definitely remain behind. On the other hand, it also seems reasonable to argue that since one of Bob or Carl had to stay in jail, this new information could not possibly change the odds for Alice. In fact, either answer may be correct. It depends on the strategy that the jailer follows in deciding which name to give to Alice (assuming that Alice knows which strategy is being used). If the jailer picks a name at random whenever both Bob and Carl are possible choices, then Alice's chances of freedom have not changed. However, if the jailer names Bob whenever possi- ble, and otherwise names Carl, then the new information does indeed change Alice's probability of release to .See Exercise 5.26 There are many other versions of the three prisoners problem, including the "Monty Hall problem" that is a staple of popular culture. Exercise 5.27 describes the Monty Hall problem and other fun applications of these ideas.

Answer 1

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