<s>[INST] Expected number of games in a best of 7 series 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.
 [/INST] 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*}
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