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October 6, 2005

Crooked Numbers

Crappy Odds

by James Click

It’s a crapshoot.

That’s the easy way to explain the playoffs, but more so since the advent of divisional play and Billy Beane’s now famous line, “My [means of building a baseball team by continually taking advantage of tiny inefficiencies in the market] doesn’t work in the playoffs.” Saying that the playoffs are a crapshoot is the easy way of saying that small sample sizes render the minute differences between playoffs teams largely moot. But largely does not equal entirely.

Take the Cardinals and Padres series. Assuming that each team’s true probability of winning a baseball game is their winning percentage for the season, that gives the Cardinals a .617 percentage and the Padres a .506 chance. Matching those two teams, we can estimate the Cardinals’ chances of winning the game by a shorthand version of the log5 method as .500 + .617 - .506 = .611. Conversely, the Padres’ odds of winning a single game between the two teams are .389. This doesn’t mean that the Cardinals have a 61% chance of winning the series for several reasons.

The primary reason is that it’s not a one-game playoff, but rather a five game series. To estimate the chance of the Cardinals winning the series, we can use a method similar to Pascal’s Triangle or Khayyam-Pascal’s Triangle or n choose k. However, because the series ends as soon as one team wins three games, we’ll use a slightly different arrangement. Assuming that the Cardinals have the same odds of winning each game, the odds of them sweeping the Padres before the series begins is (.611)^3 = 22.8%. Because there are three ways for them to win three of four (win-win-loss-win, win-loss-win-win, loss-win-win-win), the odds that they will win three of four is 3 * (.611)^3 * (1-.611) = 26.6%. The odds of winning in five games is similar: 6 * (.611)^3 * (1-.611)^2 = 20.7%. Adding those three numbers up gives the Cardinals a 70.1% chance of winning the series. While their chances of winning a five-game series are higher than their chances of winning a single game over the Padres, the Redbirds will still lose nearly three of every ten times they play a five games series.

The second reason the Cardinals likely don’t have a 70% chance of winning each game is because of home field advantage. Home teams win about 54% of their games over the course of the season, so that gives the Cardinals a .660 winning percentage at home and .562 on the road. The calculation to determine the Cardinals’ odds of winning the series gets rather complicated now because there are both home winning percentages and road winning percentages, but in the end, St. Louis comes out with a 71.8% chance to take the series, compared to their 70.1% chance when ignoring home field advantage. Intuitively, this seems about right because the Cards only get a single extra game at home. Assuming that additional 1.7% chance to win the series is similar in other series--we’ll get to those in a minute--all the hand-wringing going on in the AL over Buck Showalter pulling his starters against the Angels in the last game of the season seems a little silly. That’s not to say that a 1.7% chance is worthless, but only once in every 59 series will that make the difference.