2006 iTest Problems/Problem 23
Problem
Jack and Jill are playing a chance game. They take turns alternately rolling a fair six sided die labeled with the integers 1 through 6 as usual (fair meaning the numbers appear with equal probability.) Jack wins if a prime number appears when he rolls, while Jill wins if when she rolls a number greater than 1 appears. The game terminates as soon as one of them has won. If Jack rolls first in a game, then the probability of that Jill wins the game can be expressed as where and are relatively prime positive integers. Compute .
Solution
The probability of Jack winning in one round is , while the probability of Jill winning in one round is . In order for Jill to win, Jack must not roll winning conditions, while Jill must roll a winning condition.
If Jill wins the game in round 2, the probability is . If Jill wins the game in round 4, she must fail to roll winning conditions in round 2, so the probability is . With similar reasoning, we can find the probability where Jill wins the game in a certain round. Since all cases are mutually exclusive, we can let be the probability of Jill winning and add all of the probabilities.
The expression is an infinite geometric series with the common ratio between 0 and 1, so we can use the infinite geometric series formula.
The probability of Jill winning is , so .
See Also
2006 iTest (Problems, Answer Key) | ||
Preceded by: Problem 22 |
Followed by: Problem 24 | |
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