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.
My solution:
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!
Best Answer
Since we know the series lasted only five games, there are four possible outcomes:
$$ 1. LWWWW\\ 2. WLWWW\\ 3. WWLWW\\ 4. WWWLW $$
Thus, the probability that they Team A lost game one is $1/4$.