Problem:
Zou and Chou are practicing their -meter sprints by running races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is if they won the previous race but only if they lost the previous race. The probability that Zou will win exactly of the races is , where and are relatively prime positive integers. Find .
Solution:
Zou will win exactly 5 out of 6 races if her record over the last 5 races is WWWWL, WWWLW, WWLWW, WLWWW, or LWWWW, where W represents a race won and L represents a race lost. In the first case, the sequence of winners changes once, and in the other four cases, the sequence of winners changes twice. Thus the probability that one of these sequences occurs is
The requested sum is .