Two equally strong tennis players play against each other until one of them wins three matches in a row. What is the coefficient of variation of the number of matches played.
Can we assume that the $p$ of winning three in a row is $\left(\frac{1}{2}\right)^3$,and not winning is $1-\left(\frac{1}{2}\right)^3$, and treat this as geometric .
So the mean becomes $\frac{1-p}{p}$?
Unable to solve further
Yes you can solve through geometric distribution but you have to look into how it works for n successes or failures in a row.
Here we are seeking 3 consecutive successes (player 1 wins 3) or 3 consecutive failures (player 1 loses 3) with each success chance being $\frac{1}{2}$ and each failure chance also being $\frac{1}{2}$. Assume player 1 winning a match is success and player 1 losing (player 2 winning) is a failure. We are assuming that outcome of a match is independent of the previous outcomes. It is similar to a fair coin being tossed and you are seeking 3 consecutive heads or tails.
Expected number of matches before you get 3 consecutive successes will be $ \frac{1 + \frac{1}{2} + (\frac{1}{2})^2}{(\frac{1}{2})^3}$. This translates to 14.
It is going to be the same for 3 consecutive failures (other player winning).
But as we are fine with either 3 consecutive successes or 3 consecutive failures, it will translate to 7 matches.
You may also want to read through Markov chain.