2005 AMC 10A Problems/Problem 18

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?

Solution

There are at most games played.

If team won the first two games, team would need to win the next three games. So the only possible order of wins is .

If team won the first game, and team won the second game, the possible orders of wins are , , and , where denotes that the th game wasn't played.

There is possibility where team wins the first game, and total possibilities when team wins the series and team wins the second game. Note that the fourth possibility, , occurs twice as often as the others because it is dependent on the outcome of only games rather than , so we put over total possibilities. The desired probability is then .

Note

The original final problem was poorly worded, since the problem directly stated that the answer is (the probability of team simply winning the first game, as opposed to the desired conditional probability of this event given that both team wins the second game and team wins the series).

The problem should therefore say something like "What fraction of possible sets of game outcomes have winning the first game?" or "Given the observed results, what is the conditional probability that won the first game?"

(Many problems in probability are poorly worded.)

See Also

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.