91Ó°ÊÓ

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

Two cards are randomly selected from a deck of 52 playing cards. (a) What is the probability they constitute a pair (that is, that they are of the same denomination)? (b) What is the conditional probability they constitute a pair given that they are of different suits?

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: $$ \begin{aligned} &E_{1}=\\{\text { the first pile has exactly } 1 \text { ace }\\} \\ &E_{2}=\\{\text { the second pile has exactly } 1 \mathrm{ace}\\} \\ &E_{3}=\\{\text { the third pile has exactly } 1 \text { ace }\\} \end{aligned} $$ \(E_{4}=\\{\) the fourth pile has exactly 1 ace\\}

If the occurrence of \(B\) makes \(A\) more likely, does the occurrence of \(A\) make \(B\) more likely?

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) \(F \subset E ?\)

Bill and George go target shooting together. Both shoot at a target at the same time. Suppose Bill hits the target with probability \(0.7\), whereas George, independently, hits the target with probability \(0.4 .\) (a) Given that exactly one shot hit the target, what is the probability that it was George's shot? (b) Given that the target is hit, what is the probability that George hit it?

What is the conditional probability that the first die is six given that the sum of the dice is seven?

Suppose all \(n\) men at a party throw their hats in the center of the room. Each man then randomly selects a hat. Show that the probability that none of the \(n\) men selects his own hat is $$ \frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}-+\cdots \frac{(-1)^{n}}{n !} $$ Note that as \(n \rightarrow \infty\) this converges to \(e^{-1}\). Is this surprising?

A fair coin is continually flipped. What is the probability that the first four flips are (a) \(H, H, H, H ?\) (b) \(T, H, H, H ?\) (c) What is the probability that the pattern \(T, H, H, H\) occurs before the pattern \(H, H, H, H ?\)

Consider two boxes, one containing one black and one white marble, the other, two black and one white marble. A box is selected at random and a marble is drawn at random from the selected box. What is the probability that the marble is black?