In the World Series of baseball, two teams A and B play a sequence of games against each other, and the first team that wins a total of four games becomes the winner
of the World Series. If the probability that team A will win any particular game against team B is 1/3, what is the probability that team A will win the World Series?
I can solve this easily my way, by summing the probabilities of winning on the 4th, 5th, 6th, and 7th game. My textbook offers a different solution and I'm trying to understand why it works. They say that
$\sum \matrix{7 \\ i=4} \pmatrix{7 \\ i} \left(\displaystyle\frac{1}{3}\right)^i \left(\displaystyle\frac{2}{3}\right)^{7-i} $
Would anyone kindly explain why that is a valid approach?
Best Answer
This problem can be formulated as the equivalent one:
Team A wins the majority of 7 games
Thus, you compute the probability of winning 4, 5, 6 or 7 games out of 7. This yields the desired result.