What is the probability of a world series ending after 4 games? After 5 games? 6 games? 7 games?

Assume that each team has equal probability of winning each game, regardless of who has won the previous games.

(For the baseball-challenged, the World Series ends after one team has won 4 games. Ties are not possible.)

The chances of the series ending after four games is equal to the chances of Team A winning four games in a row, plus the chances of team B winning four games in a row: 1/(2^4) + 1/(2^4) = 1/16 + 1/16 = 2/16 = 1/8.

The chances of the series being over after 5 games are twice the chanes that one of the teams will win four games and lose one in a 5-game series, not counting the case where it is the fifth game that they lose. For any series of five games, the chance of a specific sequence of wins and losses is 1/(2^5). There are four positions where the lost game can go, so the chances of each team winning for the fourth time during the fifth game is 4/(2^5). The Chances of either team doing so is 8/(2^5) = 8/32 = 1/4

The cases for 6 and 7 games is similar to that for 5 games, except that there are more losing games to distribute.

For 6 games you wind up with: 2([C(5,2)]/[2^6]) = 2(10/64) = 5/16

For 7 games you get 2([C(6,3)]/[2^7]) = 2(20/128) = 5/16

As a check, the four chances shoul add up to 1: 1/8 + 1/4 +5/8 +5/8 = (2 + 4 + 5 + 5)/16 = 16/16 = 1

Posted by TomM on 2002-10-30 04:29:04