Problem:
Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability , and each of the other five sides has probability . Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes, the probability that the third roll will also be a six is , where and are relatively prime positive integers. Find .
Solution:
The conditional probability that the third roll will be a six given that the first two rolls are sixes is the conditional probability that Charles rolls three sixes given that his first two rolls are sixes. This is
The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions