World Series

The Problem

The Red Sox and the Yankees are playing a best-of-7 series: the first team to 4 wins takes the trophy. The Red Sox win each game independently with probability $p$.

Questions

1. Let $P(i,j)$ be the probability the Red Sox win the series given they have $i$ wins and the Yankees have $j$ wins. Write the recurrence and base cases.
2. Compute $P(0,0)$ for $p=\frac{1}{2}$ and for $p=0.55$.