<p>Zou and Chou are practicing their <img alt="$100$" height="13" src="https://latex.artofproblemsolving.com/e/5/9/e59e2c6e83eb78cca610a5fd4070ae01c8d4ae60.png" width="26" />-meter sprints by running <img alt="$6$" height="12" src="https://latex.artofproblemsolving.com/6/0/1/601a7806cbfad68196c43a4665871f8c3186802e.png" width="8" /> races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is <img alt="$\frac23$" height="37" src="https://latex.artofproblemsolving.com/6/1/1/611d08a8c57a214b2b964bba16da8157a5b81c9a.png" width="11" /> if they won the previous race but only <img alt="$\frac13$" height="37" src="https://latex.artofproblemsolving.com/e/b/a/ebafcc60a28d8759185d7808c60ea53efc73938e.png" width="11" /> if they lost the previous race. The probability that Zou will win exactly <img alt="$5$" height="12" src="https://latex.artofproblemsolving.com/7/9/0/79069377f91364c2f87a64e5f9f562a091c8a6c1.png" width="8" /> of the <img alt="$6$" height="12" src="https://latex.artofproblemsolving.com/6/0/1/601a7806cbfad68196c43a4665871f8c3186802e.png" width="8" /> races is <img alt="$\frac mn$" height="33" src="https://latex.artofproblemsolving.com/9/b/9/9b9ffa9646e74f6e94a4a66b824c3a3f08f4f4b9.png" width="18" />, where <img alt="$m$" height="8" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="15" /> and <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> are relatively prime positive integers. Find <img alt="$m+n.$" height="12" src="https://latex.artofproblemsolving.com/5/d/b/5db26f75f6218a382840f0ed34f5d36bcff61d33.png" width="52" /></p>

Question No: 378

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 $\frac23$ if they won the previous race but only $\frac13$ if they lost the previous race. The probability that Zou will win exactly $5$ of the $6$ races is $\frac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

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Tags: Combinatorics Probability

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