91Ó°ÊÓ

Two athletic teams play a series of games; the first team to win 4 games is declared the overall winner. Suppose that one of the teams is stronger than the other and wins each game with probability \(.6,\) independently of the outcomes of the other games. Find the probability, for \(i=4,5,6,7,\) that the stronger team wins the series in exactly \(i\) games. Compare the probability that the stronger team wins with the probability that it would win a 2-outof-3 series.

A fair coin is continually flipped until heads appears for the 10 th time. Let \(X\) denote the number of tails that occur. Compute the probability mass function of \(X .\)

There are three highways in the county. The number of daily accidents that occur on these highways are Poisson random variables with respective parameters. \(3, .5,\) and \(.7 .\) Find the expected number of accidents that will happen on any of these highways today.

Suppose that 10 balls are put into 5 boxes, with each ball independently being put in box \(i\) with probability \(p_{i}, \sum_{i=1}^{5} p_{i}=1\) (a) Find the expected number of boxes that do not have any balls. (b) Find the expected number of boxes that have exactly 1 ball.

There are \(k\) types of coupons. Independently of the types of previously collected coupons, each new coupon collected is of type \(i\) with probability \(p_{i}, \quad \sum_{i=1}^{k} p_{i}=1\) If \(n\) coupons are collected, find the expected number of distinct types that appear in this set. (That is, find the expected number of types of coupons that appear at least once in the set of \(n\) coupons.)