Expected number of games in a best of 7 series

Question

Assuming each team has a 0.5 probability of winning. Is there an easier way to do it other than bashing through the probabilities that the series runs 4,5,6,7 games.

Answer 1

There are $2 \binom{3}{0}$ ways for the game to finish after $4$ rounds.

There are $2 \binom{4}{1}$ ways for the game to finish after $5$ rounds.

There are $2 \binom{5}{2}$ ways for the game to finish after $6$ rounds.

There are $2 \binom{6}{3}$ ways for the game to finish after $7$ rounds.

Quick sanity check \begin{eqnarray*} 2 \left( \frac{1}{16} +\frac{4}{32} + \frac{10}{64} +\frac{20}{128} \right) = 1. \end{eqnarray*} Right so the expected value is \begin{eqnarray*} 2 \left( 4\frac{1}{16} +5\frac{4}{32} + 6\frac{10}{64} +7\frac{20}{128} \right) = \color{red}{5 \frac{13}{16}}. \end{eqnarray*}