Suppose that two teams play a series of games that ends when one of them has won 2 games. Suppose that each game played is, independently, won by team A with probability $p$. Let $N$ be the total number of games played. Find $\mathbb{E}[N]$ and $\mathbb{Var}(N)$.
Can someone please tell me if I'm on the right track? Feel free to destroy me.
There can be either just two or three games total (of course). (No ties allowed, I presume, otherwise you would have had to give us the probability of a tie. Moreover, you imply just the two teams, otherwise you would have had to give us the probability for the other team winning.)
The terminal sequences are: $AA, BB, ABA, ABB, BAB, BAA$. Calculate the probability of each sequence and multiply by its length and add up:
${\cal E}[n] = p^2 \cdot 2 + (1-p)^2 \cdot 2 + 2 \cdot p^2 (1-p) \cdot 2 + 2 (1-p)^2 p \cdot 2$
Can you calculate the variance, ${\cal V}[n]$? There are just six terms (of course).