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Chapter 3: Problem 7
The king comes from a family of 2 children. What is the probability that the other child is his sister?
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If \(A\) flips \(n+1\) and \(B\) flips \(n\) fair coins, show that the probability that \(A\) gets more heads than \(B\) is \(\frac{1}{2}\). HINT: Condition on which player has more heads after each has flipped \(n\) coins. (There are three possibilities.)
Show that $$ \frac{P(H \mid E)}{P(G \mid E)}=\frac{P(H)}{P(G)} \frac{P(E \mid H)}{P(E \mid G)} $$ Suppose that before observing new evidence the hypothesis \(H\) is three times as likely to be true as is the hypothesis \(G\). If the new evidence is twice as likely when \(G\) is true than it is when \(H\) is true, which hypothesis is more likely after the evidence has been observed?
Suppose that you are gambling against an infinitely rich adversary and at each stage you either win or lose 1 unit with respective probabilities \(p\) and \(1-p .\) Show that the probability that you eventually go broke is $$ \begin{array}{cl} 1 & \text { if } p \leq \frac{1}{2} \\ (q / p)^{i} & \text { if } p>\frac{1}{2} \end{array} $$ where \(q=1-p\) and where \(i\) is your initial fortune.
A simplified model for the movement of the price of a stock supposes that on each day the stock's price either moves up 1 unit with probability \(p\) or it moves down 1 unit with probability \(1-p\). The changes on different days are assumed to be independent. (a) What is the probability that after 2 days the stock will be at its original price? (b) What is the probability that after 3 days the stock's price will have increased by 1 unit? (c) Given that after 3 days the stock's price has increased by 1 unit, what is the probability that it went up on the first day?
Suppose that \(\left\\{E_{n}, n \geq 1\right\\}\) and \(\left\\{F_{n}, n \geq 1\right\\}\) are increasing sequences of events having limits \(E\) and \(F\). Show that if \(E_{n}\) is independent of \(F_{n}\) for all \(n\), then \(E\) is independent of \(F\).
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