There is a sporting event where team A has 1/3 chance of winning and team B has a 2/3 chance of winning. In order to win a team needs to be the first to win 4 matches. There are no ties.
The question: What is the probability that team A will win?
It will take no more than 7 matches for team A or B to win.
Neither team will win within 3 matches, so the order for the first 3 matches does not matter.
The possible scores where A wins are
So I need to find the binomial probability for each possibility and sum them. However, the order of the first 3 matches does not affect the outcome, so they should not be included in the nCr part.
That's where I'm stuck as to how I need to write out nCr for each possibility whilst taking into account that the order of the first 3 matches does not matter.