91Ó°ÊÓ

If two fair dice are tossed, what is the probability that the sum is \(i, i=2,3, \ldots, 12 ?\)

Argue that \(E=E F \cup E F^{c}, E \cup F=E \cup F E^{c}\).

Suppose that 5 percent of men and \(0.25\) percent of women are color-blind. A colorblind person is chosen at random. What is the probability of this person being male? Assume that there is an equal number of males and females.

A deck of 52 playing cards, containing all 4 aces, is randomly divided into 4 piles of 13 cards each. Define events \(E_{1}, E_{2}, E_{3}\), and \(E_{4}\) as follows: \(E_{1}=\\{\) the first pile has exactly 1 ace \(\\}\), \(E_{2}=\\{\) the second pile has exactly 1 ace \(\\}\), \(E_{3}=\\{\) the third pile has exactly 1 ace \(\\}\), \(E_{4}=\\{\) the fourth pile has exactly 1 ace \(\\}\) Use Exercise 23 to find \(P\left(E_{1} E_{2} E_{3} E_{4}\right)\), the probability that each pile has an ace.

Suppose that \(P(E)=0.6 .\) What can you say about \(P(E \mid F)\) when (a) \(E\) and \(F\) are mutually exclusive? (b) \(E \subset F ?\) (c) \(F \subset E ?\)

Urn 1 contains two white balls and one black ball, while urn 2 contains one white ball and five black balls. One ball is drawn at random from urn 1 and placed in an urn 2\. A ball is then drawn from urn 2. It happens to be white. What is the probability that the transferred ball was white?

Stores \(A, B\), and \(C\) have 50,75, and 100 employees, and, respectively, 50,60, and 70 percent of these are women. Resignations are equally likely among all employees, regardless of sex. One employee resigns and this is a woman. What is the probability that she works in-store \(C\)?

(a) A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random, and when he flips it, it shows heads. What is the probability that it is a fair coin? (b) Suppose that he flips the same coin a second time and again it shows heads. Now, what is the probability that it is a fair coin? (c) Suppose that he flips the same coin a third time and it shows tails. Now, what is the probability that it is a fair coin?

Three prisoners are informed by their jailer that one of them has been chosen at random to be executed, and the other two are to be freed. Prisoner \(A\) asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information, since he already knows that at least one will go free. The jailer refuses to answer this question, pointing out that if \(A\) knew which of his fellows were to be set free, then his own probability of being executed would rise from \(\frac{1}{3}\) to \(\frac{1}{2}\), since he would then be one of two prisoners. What do you think of the jailer's reasoning?

For a fixed event \(B\), show that the collection \(P(A \mid B)\), defined for all events \(A\), satisfies the three conditions for a probability. Conclude from this that $$ P(A \mid B)=P(A \mid B C) P(C \mid B)+P\left(A \mid B C^{c}\right) P\left(C^{C} \mid B\right) $$ Then directly verify the preceding equation.