2005 AMC 10A Problems/Problem 18
Contents
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
2005 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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All AMC 10 Problems and Solutions |
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