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Data indicate that the number of traffic accidents in Berkeley on a rainy day is a Poisson random variable with mean 9, whereas on a dry day it is a Poisson random variable with mean 3. Let \(X\) denote the number of traffic accidents tomorrow. If it will rain tomorrow with probability . 6 , find (a) \(E[X] ;\) (b) \(P[X=0\\}\); (c) \(\operatorname{Var}(X)\)

Let \(A\) and \(B\) be mutually exclusive events of an experiment. If independent replications of the experiment are continually performed, what is the probability that \(A\) occurs before \(B ?\)

\(A, B\), and \(C\) are evenly matched tennis players. Initially \(A\) and \(B\) play a set, and the winner then plays \(C\). This continues, with the winner always playing the waiting player, until one of the players has won two sets in a row. That player is then declared the overall winner. Find the probability that \(A\) is the overall winner.

A and \(B\) roll a pair of dice in turn, with \(A\) rolling first. \(A\) 's objective is to obtain a sum of 6 , and \(B\) 's is to obtain a sum of 7 . The game ends when either player reaches his or her objective, and that player is declared the winner. (a) Find the probability that \(A\) is the winner. (b) Find the expected number of rolls of the dice. (c) Find the variance of the number of rolls of the dice.

The opponents of soccer team \(\mathrm{A}\) are of two types: either they are a class 1 or a class. 2 team. The number of goals team A scores against a class \(i\) opponent is a Poisson random variable with mean \(\lambda_{i}\), where \(\lambda_{1}=2, \lambda_{2}=3\). This weekend the team has two games against teams they are not very familiar with. Assuming that the first team they play is a class 1 team with probability \(0.6\) and the second is, independently of the class of the first team, a class 1 team with probability \(0.3\), determine (a) the expected number of goals team \(\mathrm{A}\) will score this weekend. (b) the probability that team \(\mathrm{A}\) will score a total of five goals.