A series that goes the distance
Two equally strong teams play a best-of-seven series: the first to win 4 games wins. Each game is won by either team with probability , independently.
Find the probability that a seventh game is played.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
A seventh game is played exactly when nobody has won after six games.
What must the score be after six games?
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Game 7 is needed exactly when the first six games end 3 to 3. To count that, imagine all six games are played even if the series is decided earlier: that changes nothing about whether it is 3 to 3.
Step 1. After six games, nobody has 4 wins exactly when the score is 3 to 3.
Step 2. If all six games are played, the number of wins of the first team is binomial with and , so (If one team had 4 wins earlier, the score after six imagined games could not be 3 to 3, so the extended games do not change the event.)
So a seventh game is played with probability .