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In this exercise we will use Bayes’ theorem to solve the Monty Hall puzzle (Example 10 in Section 7.1). Recall that in this puzzle you are asked to select one of three doors to open. There is a large prize behind one of the three doors and the other two doors are losers. After you select a door, Monty Hall opens one of the two doors you did not select that he knows is a losing door, selecting at random if both are losing doors. Monty asks you whether you would like to switch doors. Suppose that the three doors in the puzzle are labeled 1, 2, and 3. Let W be the random variable whose value is the number of the winning door; assume that p(W = k) = 1?3 for k = 1, 2, 3. Let M denote the random variable whose value is the number of the door that Monty opens. Suppose you choose door i. a) What is the probability that you will win the prize if the game ends without Monty asking you whether you want to change doors? b) Find p(M = j ? W = k) for j = 1, 2, 3 and k = 1, 2, 3. c) Use Bayes’ theorem to find p(W = j ? M = k) where i and j and k are distinct values. d) Explain why the answer to part (c) tells you whether you should change doors when Monty gives you the chance to do so.

Let X and Y be the random variables that count the number of heads and the number of tails that come up when two fair coins are flipped. Show that X and Y are not independent.

Show that if \(E\) and \(F\) are independent events, then \(\overline{E}\) and \(\overline{F}\) are also independent events.

Prove Theorem \(2,\) the extended form of Bayes' theorem. That is, suppose that \(E\) is an event from a sample space \(S\) and that \(F_{1}, F_{2}, \ldots, F_{n}\) are mutually exclusive events such that \(\bigcup_{i=1}^{n} F_{i}=S .\) Assume that \(p(E) \neq 0\) and \(p\left(F_{i}\right) \neq 0\) for \(i=1,2, \ldots, n .\) Show that $$ p\left(F_{j} | E\right)=\frac{p\left(E | F_{j}\right) p\left(F_{j}\right)}{\sum_{i=1}^{n} p\left(E | F_{i}\right) p\left(F_{i}\right)} $$ \(\left[\text {Hint} : \text { Use the fact that } E=\bigcup_{i=1}^{n}\left(E \cap F_{i}\right) .\right]\)

If \(E\) and \(F\) are independent events, prove or disprove that \(\overline{E}\) and \(F\) are necessarily independent events.

In Exercises 18, 20, and 21 assume that the year has 366 days and all birthdays are equally likely. In Exercise 19 assume it is equally likely that a person is born in any given month of the year. a) What is the probability that two people chosen at random were born during the same month of the year? b) What is the probability that in a group of n people chosen at random, there are at least two born in the same month of the year? c) How many people chosen at random are needed to make the probability greater than 1?2 that there are at least two people born in the same month of the year?

Let \(X\) be the number appearing on the first die when two fair dice are rolled and let \(Y\) be the sum of the numbers appearing on the two dice. Show that \(E(X) E(Y) \neq E(X Y) .\)

What is the probability that a five-card poker hand contains a royal flush, that is, the \(10,\) jack, queen, king, and ace of one suit?

What is the expected value of the sum of the numbers appearing on two fair dice when they are rolled given that the sum of these numbers is at least nine. That is, what is \(E(X | A)\) where \(X\) is the sum of the numbers appearing on the two dice and \(A\) is the event that \(X \geq 9 ?\)

February 29 occurs only in leap years. Years divisible by 4, but not by 100, are always leap years. Years divisible by 100, but not by 400, are not leap years, but years divisible by 400 are leap years. a) What probability distribution for birthdays should be used to reflect how often February 29 occurs? b) Using the probability distribution from part (a), what is the probability that in a group of n people at least two have the same birthday?