## What Is the Probability of Winning Back-to-Back Baseball Games?

At Decision Science News, Dan Goldstein asks the question, looks at the data, and finds a surprising result.

If a team wins on one day, what’s the probability they’ll win against the same opponent when they play the very next day?

We asked two colleagues knowledgeable in baseball and the mathematics of forecasting. The answers came in between 65% and 70%.

The true answer: 51.3%, a little better than a coin toss.

That’s right. When you win in baseball, there’s only a 51% chance you’ll win again in more or less identical circumstances. The careful reader might notice that the answer is visible in the already mentioned chart. The reversals of size 0, (meaning no reversal, meaning the same team won twice) occur 51,296 times per 100,000 pairs of consecutive games.

Statistician Andrew Gelman is not surprised and explains why.

I have to say, I’m surprised his colleagues gave such extreme guesses. I was guessing something like 50%, myself, based on the following very crude reasoning:

Suppose two unequal teams are playing, and the chance of team A beating team B is 55%. (This seems like a reasonable average of all matchups, which will include some more extreme disparities but also many more equal contests.) Then the chance of the same team winning both games is .55^2 + .45^2 = .505. Even .6^2 + .4^2 is only .52.

Two interesting posts, worth reading. Thanks to Jonathan for the pointer.

### 4 Responses “What Is the Probability of Winning Back-to-Back Baseball Games?”

1. Jonathan says:

Thanks, JC. There’s an interesting psychological effect at work here, I think. One naturally tends to think that the better team won the first game, which then makes it very likely they win the second game. But of course 45 (or 40 or whatever) percent of the time the worse team wins the first game. You certainly don’t think that makes it likely to win the second. That, and the fact that 70-30 matchups are pretty rare. The other psychological factor is the false “hot-hand” hypothesis, a la Gilovich.

2. Zach says:

I don’t see why this is surprising at all. Sporting events are a zero-sum game (each game has exactly one winner and one loser). When aggregated across all games for all teams, then it should be clear that the probability would be 50%.

3. John Beamer says:

Zach — depends on the context right. In basketball if Team A beat Team B then your Bayesian prior would probably be that team A is around a 60-65% team. However, in baseball random variation is much more important so your prior is closer to 50% (depending on what else you knew i.e., if you knew whether it was the home team or road team that won. So in the context of baseball it isn’t surprising. But it might be if the academics questioned knew nothing about baseball. It is why being a genuine SME (rather than a casual fan) is important when interpreting data.