91Ó°ÊÓ

Poker dice is played by simultaneously rolling 5 dice. Show that (a) \(P\\{\text { no two alike }\\}=.0926\) (b) \(P\\{\text { one pair }\\}=.4630\) (c) \(P\\{\text { two pair }\\}=.2315\) (d) \(P\\{\text { three alike }\\}=.1543\) (e) \(P\\{\text { full house }\\}=.0386\) (f) \(P\\{\text { four alike }\\}=.0193\) (g) \(P\\{\text { five alike }\\}=.0008\)

If 8 rooks (castles) are randomly placed on a chess-board, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook.

Two symmetric dice have had two of their sides painted red, two painted black, one painted yellow, and the other painted white. When this pair of dice is rolled, what is the probability that both dice land with the same color face up?

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

If two dice are rolled, what is the probability that the sum of the upturned faces equals \(i ?\) Find it for \(i=\) \(2,3, \ldots, 11,12\)

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 \(11,\) the player wins. If the outcome is anything else, the player continues to roll the dice until 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 appears, 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 \(E_{i, n}\) 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 5 red, 6 blue, and 8 green balls. If a set of 3 balls is randomly selected, what is the probability that each of the balls will be (a) of the same color? (b) of different colors? Repeat under the assumption that whenever a ball is selected, its color is noted and it is then replaced in the urn before the next selection. This is known as sampling with replacement.

A 3 -person basketball team consists of a guard, a forward, and a center. (a) If a person is chosen at random from each of three different such teams, what is the probability of selecting a complete team? (b) What is the probability that all 3 players selected play the same position?

The second Earl of Yarborough is reported to have bet at odds of 1000 to 1 that a bridge hand of 13 cards would contain at least one card that is ten or higher. (By ten or higher we mean that a card is either a ten, a jack, a queen, a king, or an ace.) Nowadays, we call a hand that has no cards higher than 9 a Yarborough. What is the probability that a randomly selected bridge hand is a Yarborough?