91Ó°ÊÓ

A pair of fair dice are rolled. What is the probability that the second die lands on a higher value than does the first?

A pair of dice is rolled until a sum of either 5 or 7 appears. Find the probability that a 5 occurs first. HINT: Let \(E_{n}\) denote the event that a 5 occurs on the \(n\)th roll and no 5 or 7 occurs on the first \(n-1\) rolls. Compute \(P\left(E_{n}\right)\) and argue that \(\sum_{n=1}^{\infty} P\left(E_{n}\right)\) is the desired probability.

The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a 2,3 , or 12 , the player loses; if the sum is either a 7 or an 11 , he or she wins. If the outcome is anything else, the player continues to roll the dice until he or she rolls either the initial outcome or a 7. If the 7 comes first, the player loses; whereas if the initial outcome reoccurs before the 7, the player wins. Compute the probability of a player winning at craps. HinT: Let \(E_{i}\) denote the event that the initial outcome is \(i\) and the player wins. The desired probability is \(\sum_{i=2}^{12} P\left(E_{i}\right) .\) To compute \(P\left(E_{i}\right)\), define the events \(E_{L, n}\) to be the event that the initial sum is \(i\) and the player wins on the \(n\)th roll. Argue that \(P\left(E_{i}\right)=\sum_{n=1}^{\infty} P\left(E_{i, n}\right)\).

An urn contains \(n\) white and \(m\) black balls, where \(n\) and \(m\) are positive numbers. (a) If two balls are randomly withdrawn, what is the probability that they are the same color? (b) If a ball is randomly withdrawn and then replaced before the second one is drawn, what is the probability that the withdrawn balls are the same color? (c) Show that the probability in part (b) is always larger than the one in part (a).

A forest contains 20 elk, of which 5 are captured, tagged, and then released. A certain time later 4 of the 20 elk are captured. What is the probability that 2 of these 4 have been tagged? What assumptions are you making?

There are 30 psychiatrists and 24 psychologists attending a certain conference. Three of these 54 people are randomly chosen to take part in a panel discussion. What is the probability that at least one psychologist is chosen?

Two cards are chosen at random from a deck of 52 playing cards. What is the probability that they (a) are both aces; (b) have the same value?

There are 5 hotels in a certain town. If 3 people check into hotels in a day, what is the probability they each check into a different hotel? What assumptions are you making?

A town contains 4 people that repair televisions. If 4 sets break down, what is the probability that exactly \(i\) of the repairers are called? Solve the problem for \(i=1,2,3,4\). What assumptions are you making?

Two dice are thrown \(n\) times in succession. Compute the probability that double 6 appears at least once. How large need \(n\) be to make this probability at least \(\frac{1}{2}\) ?