Problem:
Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is . One marble is taken out of each box randomly. The probability that both marbles are black is , and the probability that both marbles are white is , where and are relatively prime positive integers. What is
Solution:
Suppose that the first box has marbles, of which are black, and that the second box has marbles, of which are black. Without loss of generality, assume that . Thus and , so . The latter equation implies that divides either or . Because , both and must be divisible by . One possibility is that and , in which case , so and , because and . In this case, the probability of obtaining two white marbles is . The only other possibility is that and , in which case , so and must each be . The probability of obtaining two white marbles is in this case too. Hence .
The problems on this page are the property of the MAA's American Mathematics Competitions