My exam had the following prompt:
A best-of-seven playoff is a competition between two teams head-to-head which must win four games to win the series. Four is chosen as it would constitute a majority of games played; whoever has won four games before all seven games have been played, all other games are omitted. Note that NBA finals are played based on best-of seven games series.
We have two competing teams in a best-of-seven games series: Team A and Team B. The probability of Team A winning a game is $p$, and Team B winning a game is $1-p$ (no draw games) where $0 < p < 1$.
Hint: The winner of the series has to win the last game.
One of the questions is based on this prompt and reads:
We know that Team A won the series in five games (i.e, won 4-1). What is the probability of Team A losing the first game?
And here's how I solved it, but I only got 5/10 points for it, and I'm not sure what I did wrong exactly. The professor hasn't released the solutions yet, so I figured I'd ask here.
There is only one such outcome: LWWWW.
Probability = $(1-p)p^4$
Why is this wrong? Since the prompt says the winner of the series has to win the last game, and we're told team A wins in 5 games, we're asked to find the probability of it losing the first game under these terms. Thanks!