Dan Goldstein did an informal study asking people the following question:
When two baseball teams play each other on two consecutive days, what is the probability that the winner of the first game will be the winner of the second game?
You can make your own guess and the continue reading below.
We asked two colleagues knowledgeable in baseball and the mathematics of forecasting. The answers came in between 65% and 70%.
The true answer [based on Dan’s analysis of a database of baseball games]: 51.3%, a little better than a coin toss.
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.
Dan and his commenters discuss other factors such as home-field advantage and the quality of the starting pitcher, but I think the above reasoning basically works.
What time is it when your team loses? Time for the coach to get fired.
Dan has another question that I think I have the answer to. He writes that he “has always wondered why teams are so eager to fire their coaches after they lose a few big games. Don’t they realize that their desired state of having won those same few big games would have been mostly due to luck?”
My guess is that these are situations where the management has already decided they want to fire the coach, and they’re just waiting for a convenient time to do it so as not to antagonize the fans.