## NBA Finals (Solution)

Break the problem up into mutually exclusive outcomes:

- Warriors win in 4 games
- Warriors win in 5 games
- Warriors win in 6 games
- Warriors win in 7 games

What’s the probability that Warriors win in 4 games? That’s easy, it’s just \(0.6^4\).

What’s the probability that Warriors win in 5 games? Any given 5 game streak, such as WWLWW is equally probable, with \(P = 0.6^4 0.4^1\). How many streaks are there? The one loss can be anywhere in the first 4 games, so 4. But more generally, \({4 \choose 1} = 4\). So the probability that the warriors win in 5 games is \({4 \choose 1} 0.6^4 0.6^1\).

What’s the probability that Warriors win in 6 games? Similar to above, the two losses can come in any of the first 5 games, and each 6-game winning streak is equally likely: \({5 \choose 2} 0.6^4 0.4^2\).

You might detect a pattern, but there’s only one last case: \({6 \choose 3} 0.6^4 0.6^3\).

Putting it all together, let’s call \(L\) the number of losses and \(T\) the total number of games:

\[T = L + 4 \\ \sum_{L=0}^{3} {T - 1 \choose L} 0.6^4 0.4^L \\ \sum_{L=0}^{3} {L + 3 \choose L} 0.6^4 0.4^L \\ \approx 0.71\]