1) Fischer and Spassky play a sudden-death chess match whereby the first player to win a game wins the match.

1) Fischer and Spassky play a sudden-death chess match whereby the first player to win a game wins the match. Each game is won by Fischer with probability p, by Spassky with probability q, and is a draw with probability 1 − p − q.(a) What is the probability that Fischer wins the match? (b) What is the PMF, the mean, and the variance of the duration of the match?2) Imagine a TV game show where each contestant i spins an infinitely calibrated wheel of fortune, which assigns him/her with some real number with a value between 1 and 100. All values are equally likely and the value obtained by each contestant is independent of the value obtained by any other contestant.(a) Find P(X1 < X2). (b) Find P(X1 < X2, X1 < X3).(c) Let N be the integer-valued random variable whose value is the index of the first contestant who is assigned a smaller number than contestant 1. As an illustration, if contestant 1 obtains a smaller value than contestants 2 and 3 but contestant 4 has a smaller value than contestant 1 (X4 < X1), then N = 4. Find P(N > n) as a function of n.(d) Find E[N], assuming an infinite number of contestants.3) Alvin shops for probability books for K hours, where K is a random variable that is equally likely to be 1, 2, 3, or 4. The number of books N that he buys is random and depends on how long he shops according to the conditional PMFpN |K(n|k) = 1,