Problem:
Team and team play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team wins the second game and team wins the series, what is the probability that team wins the first game?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
There are four possible outcomes,
but they are not equally likely. This is because, in general, the probability of any specific four-game series is , whereas the probability of any specific five-game series is . Thus the first listed outcome is twice as likely as each of the other three. Let be the probability of the occurrence . Then the probability of is also , as is the probability of , whereas the probability of is . So
The only outcome in which team wins the first game is , so the probability of this occurring is .
To consider equally-likely cases, suppose that all five games are played, even if team has won the series before the fifth game. Then the possible ways that team can win the series, given that team wins the second game, are
, and ,
In only the first case does team win the first game, so the probability of this occurring is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions