Tennis Paradox Probability

Question

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.

b. What is the probability of Alice getting the $ 100 reward if she chooses the sequence CBC?

a. What is the probability of Alice getting the $ 100 reward if she chooses the sequence BCB?

I'm really having trouble figuring out where to start with this (I made a tree diagram but I don't know if it helps/what to do with it)

Thanks!

tree diagram

Answer 1

From the tree diagram you've uploaded, select the branches which yield the desired result. Then multiply the probability of that happening.

For example, the very first branch of BCB is winning against Betty, winning against Carol, and winning against Betty a second time. Clearly this satisfies the condition of winning two consecutive games, therefore this is a "success". The probability of this happening is $0.4 \times 0.5 \times 0.4$, or $0.08$ (8%).

There's a quick way to tell how to choose the order as well. Whatever happens in the other two matches, Alice must win the second match to win two consecutive games. Therefore, it's to her advantage to play the second match against the weaker opponent (Carol).