Suppose that two teams are playing a series of games, each of which is independently won by team A with probability p and by team B with probability 1-p. The winner of the series is the first team to win i games.
If i=4, find the probability that a total of 7 games are played. Also show that this probability is maximized when p=0.5
I thought that this is a pretty straight forward negative binomial problem, but I don't seem to be getting the correct answer. Here's my work:
The problem is that the probability of 7 games being played doesn't seem to be maximized when p=0.5.
For example, if I set p=0.6 then I get:
I assume that I'm reading the question wrong, using the wrong distribution, committing a math error, or some combination of the three. Any help is appreciated.