Two persons, say A and B, alternate between playing a video game against the computer. Person A has constant probability p1, 0 < p1 < 1 of winning, while person B has p2, 0 < p2 < 1. First A plays, then B, then A, etc., until someone wins against the computer. Assume that each game is independent from the previous one, i.e. there is, for example, no ‘learning effect’. Let A and B be the events that persons A and B win, respectively.

(a) Write closed form expressions for Pr(A) and Pr(B). Define qi = 1 − pi, i = 1, 2.

(b) A game between two people is said to be fair if Pr(A) = Pr(B). Give an algebraic expression for p2 in terms of p1 such that the game is fair and examine how p2 behaves for various p1.