Chapter 6, Problem 3P

First Paradox: Under certain circumstances, you have your best chance of winning a tennis

tournament if you play most of your games against the best possible opponent.

Alice and her two sisters, Betty and Carol, are avid tennis players. Betty is the best of the three sisters, and Carol plays at the same level as Alice. Alice defeats Carol 50% of the time but only defeats Betty 40% of the time.

Alice’s mother offers to give her $100 if she can win two consecutive games when playing three alternating games against her two sisters. Since the games will alternate, Alice has two possibilities for the sequence of opponents. One possibility is to play the first game against Betty, followed by a game with Carol, and then another game with Betty. We will refer to this sequence as BCB. The other possible sequence is CBC.

Calculate the probability of Alice getting the $100 reward if she chooses the sequence BCB.