# probability help, i don't get it?

Suppose that 2 sports teams (team A and team B) are equally likely to win when they play each other (so the probability of either team winning is 0.5). Suppose they are playing a 5 game series (whoever wins 3 of the 5 games wins the series...the teams stop playing after one team wins 3 games). What is the probability all 5 games will be played (or asking it another way, what is the probability that in the first four games, each team wins exactly 2)? (please round your answer to 3 decimal places)

Relevance

Note the winner of the game by the letter of the wining team.

Pr(AAA) = 1/8

P(AABA) = (ABAA) = (BAAA) = 1/16 and together the 3 of them = 3/16

Together those total 5/16 ...

Then we multiply by 2 to account for the ones where B wins

and get 10/16 = 5/8 chance that the series ends in 3 or 4 games.

Which leaves 3/8 chance that it goes the full 5 games. << ANSWER

To check:

For the series to last 5 games, the first 4 games must be divided 2 and 2.

So there are 4c2 = 6 ways to designate which two of the 4 games A wins

(leaving the others for B).

Pr( each sequence of 4 games ) = 1/16

hence for 6 of them, overall Pr = 6/16 = 3/8 which matches the previous calculation perfectly.