Fischer and Spassky play a chess match in which the first pl

Fischer and Spassky play a chess match in which the first player to win a game wins the match. After 10 successive draws, the match is declared drawn. Each game is won by Fischer with probability 0.4, is won by Spassky with probability 0.3, and is a draw with probability 0.3, independent of previous games. (a) What is the probability that Fischer wins the match? (b) What is the PMF of the duration of the match?

Solution

Answer :-

a) Fischer winning is the sum from i = 0 to 9 of (0.3^i) * (0.4) this accounts for draws until Fischer wins.
b) P(X=10) = 0.3^10, else P(X = k) = sum from k=1 to 9 of (0.3^k-1) * (0.7) where the 0.3 is the probability of k many draws happening, then 0.7 is the probability that someone wins a match.

2. Just integrate c*2^k from 1 to 3 and set that equal to 1.