Problem:
There is a chance of rain on Saturday and a chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is , where and are relatively prime positive integers. Find .
Solution:
Let be the probability that it rains on Saturday and not on Sunday, be the probability that it rains on Sunday and not on Saturday, and be the probability that it rains on both days. Then the conditions of the problem imply that , and the conditional probabilities satisfy or . Thus , so which gives . The required probability is . The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions