The Role of Chance in Baseball

Lights flash throughout the stadium. The noise of the crowd is constant, rising with the excitement, but never dipping below the steady din of the conversations of 40,000 fans. It's a scene that would make Donald Trump proud. The pitcher sets himself in the cold October night. He delivers, and the ball turns end over end, hurling toward the plate, tipping and rolling, as the 40,000 sit on edge waiting to see where the dice stop: the crack of wood is a seven, the snap of leather a two. Yes, this is playoff baseball-a wondrous game, an enthralling spectacle, a beautifully wrapped crap shoot, a pair of dice cloaked in the charms of our modern rounders.

You know the story by now. Even if you never read Moneyball, you've no doubt heard Billy Beane's famous quote about the nature of the playoffs. In a game where even the Royals can expect to beat the Yankees 3 out of 10 times (even if they never pitched Zack Greinke), it's fairly obvious how much chance can enter into the outcome of a short series. Never has this been more obvious than in 2006, when the St. Louis Cardinals established themselves as an 83 win team over the course of 161 games, and then proceeded to out-roll the more potent Padres, Mets, and Tigers in one quick crap shoot after another, riding hot rollers Jeff Suppan, et al to a World Series title.

At least that's the story you tend to hear, even from the typically savvy among us. An 83 win team cannot really be better than a 97 or a 95 win team, but, of course, when you set them at the craps table and call out, "First to four!", anything can happen. It is, of course, true that there is a lot of chance involved in the outcome of a 5- or 7-game series. The inferior team will win a pretty significant portion of the time. What is missing from this story, however, is that the 162 game season is subject to the same laws of chance. Setting n=162 rather than n=7 certainly seems like a good way to cut out the noise and establish the true talent levels of each team, but the reality is that in baseball, where even the best and the worst teams fall within the .400 to .600 range, there's still a lot of chance involved.

Let's look at the 2006 Cardinals as an example. Let's assume, just for example's sake, that the Cardinals' .516 win percentage established their actual ability, and that each team they played was as good as its 162 game record. To win the World Series, the Cardinals had to go through a 5-game set with the 88-win Padres, a 7-game set with the 97-win Mets, and a 7-game set with the 95-win Tigers. While each series on it's own might give the Cardinals a decent chance for success, they had to win all three in succession. What are their odds of getting all the way through?

To keep the math simple, we'll ignore homefield advantage and individual pitcher match-ups and just assume that each team's probability of winning each game is the same. Against the .543 Padres, the .516 Cardinals would have a .472 chance of winning each game (using the log5 method), which would mean they have a .448 chance of winning the 5-game series. That's pretty good. Now on to the Mets.

This was a taller order. Not only did the Mets win 9 more games than the Padres, they had 2 extra games to let the better team work its way to the top. Here, the Cards would have just a .416 shot at each game and a .322 shot at the series. Against the similarly strong 95-win Tigers in the Series, the Cardinals had a .347 chance at winning the set.

The combined chance of all of these events happening together is just .050. If you bump up the Tigers a few wins for their tougher AL schedule, it's just .045. That's a mere 5% chance of the Cardinals winning the World Series if you assume that each team's regular season record establishes its true strength. In other words, if that assumption is true, then the Cardinals winning the World Series is a very unlikely outcome-hardly the type of thing you would expect to be described as a crap shoot.

Regular season records don't establish a team's true strength, however. There are, of course, issues of changing personnel, such as teams having health problems throughout the year and getting healthy for the playoff run (as the Cardinals did), but that's not what I'm talking about. Just the pure random chance that goes into a team's 162-game record can't be ignored any more than the chance that goes into a playoff series can. Recall that the chances of an 83-win team beating an 88-win team, a 97-win team, and then a 95-win team in the playoffs is just 5%. What kind of variability in a team's regular season record would cover that same likelihood?

The Cardinals won 83 out of 161 games. We want to know how good a team could be to still have a 5% chance of winning only 83 out of 161 games. That would be a .583 team, or a 94.4 win team. A team whose actual ability was .583 would be expected to win no more than 83 of 161 games roughly 5% of the time.

So when you think about the role of chance in the playoffs and that the winner of the series does not necessarily determine the better team, don't forget that the same caveat covers the regular season as well. When you see something like the 83-win Cardinals winning the World Series, the role of chance in the outcome of events could mean that the Cardinals were a bad team that got lucky in the playoffs, but it could just as well mean that the Cardinals were a good team that got unlucky in the regular season. After all, the chances that a team that was truly as bad as the Cardinals' record beating 3 teams that were truly as good as the Padres', Mets', and Tigers' records in the playoffs are the same as a team that is truly as good as a 94-95 win team performing as poorly as the Cardinals did over the regular season.