Expected number of games in the World Series

Sadly the MLB season is almost over, but the NBA '14 - '15 season starts on the 28th (if the World Series goes to game 7, it will be on the 29th,) I was talking to my friends about the perfect transition from baseball seasons to basketball seasons and whether there will be game 7 and I became curious.

It is actually not too hard to calculate, assuming both teams have the same probability of winning.

1) The probability that the series ends in 4 is 1/8. The winner of the first game must win the second game (1/2,) third game (1/2,) and the fourth game (1/2.)

2) The probability that the series ends in 5 is 1/4. There are few ways to see this, but I think the easiset way is this: For the winner of the game 5 to be the winner of the series, they have to win 3 out of the first 4 games. There are 16 possible outcomes for the first four games and among those, 4 of them is what we want: 4 choose 1 ways to select the games to lose for the series winner (and win the rest.)

3) The probability that the series ends in 6 is 5/16. Similarly, we can calculate it by 5 choose 2 over 32.

4) The probability that the series ends in 7 is also 5/16, 6 choose 3 over 64.

1) ~ 4) add up to 1, which it should. Is it strange that the probabilities that the series end in 6 or 7 are the same? If you think about it, it is not too difficult to see: in order for the series to end in 6, a team must lead 3 games to 2 games. If leading team wins game 6, the series ends in 6; if other team wins game 6, the series ends in 7.

Going back to original question, the expected number of games in a best-of-seven series would be 4*1/8+5*1/4+6*5/16+7*5/16=93/16, about 5.813 games,

I looked them up. Other than 1903, 1919, 1920, 1921, every WS was best-of-seven series and there were 105 of them.

	ends in 4	ends in 5	ends in 6	ends in 7	total
theoretical	12.5%	25%	31.25%	31.25%	100%
actual	21 (20%)	24 (22.86%)	24 (22.86%)	36 (34.29%)	105 (100%)

More of game 7 than expected. Hmm.. I guess in real life if you're up three games to two, you are not as motivated as teams down two games to three. However, there were more one-sided series than expected as well, which resulted the average number of games to be just 5.714 (slightly lower than 5.813.) This might mean we are more likely to see the end of baseball season this year before the basketball season starts.

If we just take a look at the record of teams facing elimination, they are at 94 - 105 (47.23%.) However, that included game 7's, which means both teams are facing elimination and one of them must win. If we take those out, it becomes down to 58 - 69 (45.67%.) Not by much, but it is indeed hard to win games if you're facing elimination. The Red Sox was just weird/lucky when they were keep winning.

How about the expected number of games when the series is tied after playing two games? Obviously we would expect more games than 5.81 games since there are no sweeps.

Theoretically, that's the same as 2+(expected number of games in best-of-five series,) which would be 6.125. For such World Series (there were 54 of them) in the past, the distribution was the following:

	ends in 5	ends in 6	ends in 7	total
theoretical	25%	37.5%	37.5%	100%
actual	14 (25.93%)	16 (29.63%)	24 (44.44%)	54 (100%)

Again, we were able to see more of game 7. The average number of games was 6.185, which was similar to expected. If I had to bet, I think the series will end in game at the Kauffman Stadium.