Problem:
Jen enters a lottery by selecting four distinct elements of . Then four distinct elements of are drawn at random. Jen wins a prize if at least two of her numbers are drawn, and she wins the grand prize if all four of her numbers are drawn. The probability that Jen wins the grand prize given that Jen wins a prize is , where and are relatively prime positive integers. Find .
Solution:
There is only way for Jen to win the grand prize, and that is for the drawn numbers to match all four of her numbers. The number of ways to match exactly of the numbers drawn is . The number of ways to match exactly of the numbers drawn is . Thus there are equally likely ways to win a prize. The probability that Jen wins the grand prize given that Jen wins a prize is therefore . The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions