Atable tennis match is played between jack and rose. the winner of the match is the one who first wins 4 games in total, and in any game the winner is the one who first scores 11 points. note that in an individual game, if the score is 10 to 10, the game goes into extra play (called deuce) until one player has gained a lead of 2 points. let p be the probability that rose wins a point in any single round of serve, and assume that different rounds in all games are independent. (i) how many games can there be at most before a match winner appears? (ii) in a given individual game, what is the probability that the game runs into the deuce stage? (iii) suppose that p=0.6. compute the probability that rose wins the match? (iv) as a function of p, let f(p) be the probability that the match ends with a maximal number of games. for what value(s) of p is f(p) largest? justify your answer and compute the resulting probability.

When booking personal travel by air, one is always interested in actually arriving at one’s final destination even if that arrival is a bit late. the key variables we can typically try to control are the number of flight connections we have to make in route, and the amount of layover time we allow in those airports whenever we must make a connection. the key variables we have less control over are whether any particular flight will arrive at its destination late and, if late, how many minutes late it will be. for this assignment, the following necessarily-simplified assumptions describe our system of interest: the number of connections in route is a random variable with a poisson distribution, with an expected value of 1. the number of minutes of layover time allowed for each connection is based on a random variable with a poisson distribution (expected value 2) such that the allowed layover time is 15*(x+1). the probability that any particular flight segment will arrive late is a binomial distribution, with the probability of being late of 50%. if a flight arrives late, the number of minutes it is late is based on a random variable with an exponential distribution (lamda = .45) such that the minutes late (always rounded up to 10-minute values) is 10*(x+1). what is the probability of arriving at one’s final destination without having missed a connection? use excel.