Question 1
Zou and Chou are practicing their $100$-meter sprints by running $6$ races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\frac{2}{3}$ if they won the previous race but only $\frac{1}{3}$ if they lost the previous race. The probability that Zou will win exactly $5$ of the $6$ races is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$