Hot Hands and Tuesday’s Children

Image: Neon Tommy, via Flickr. CC BY-SA 2.0.

Image: Neon Tommy, via Flickr. CC BY-SA 2.0.

People, especially sports fans, seem to believe that players can get “hot,” that they will hit more baskets (or succeed in whatever metric is of interest in their sport) more after a series of hits than after a series of misses. We statistics-savvy types are far too sophisticated to believe something like that, though, which is why we call it the fallacy of the hot hand. In 1985, Thomas Gilovich, Amos Tversky, and Robert Vallone wrote a paper suggesting that the “hot hand” doesn’t exist; it’s just a misinterpretation of the natural streakiness of independent events. But a few months ago, Joshua B. Miller and Adam Sanjurjo published an article that claims to find an important flaw in the 1985 paper. Perhaps there is a hot hand after all!

I’ve read several posts and articles about the fallacy of the fallacy of the hot hand recently. Jordan Ellenberg was on the case for Slate with a good explanation of the flaw (the moral of the story: averaging ratios is a dicey prospect), and George Johnson wrote about it for the New York Times. I also enjoyed the Gödel’s Lost Letter post about the problem, complete with a suggestion for fixing it.

Andrew Gelman has the most extensive coverage of the hot hand with four posts about it in the past few months and a nice one from 2014 as well.

One thing I really like about Gelman’s writing in general is thefact that he emphasizes effect size so much. He wrote, “A better framing is to start from the position that the effects are certainly not zero. Athletes are not machines, and anything that can affect their expectations (for example, success in previous tries) should affect their performance—one way or another.” To me, this puts the hot hand question into better perspective. (On the other hand, it also makes me want to give up—how can we possibly untangle all the contributing factors to any aspect of human behavior or performance? I suppose that’s why I’m in math and not psychology.)

The problem Miller and Sanjurjo found with the earlier research boils down to a fairly simple observation about probability, so posts about it eventually spill into questions about coin flips and boys and girls born in different families. Steven Landsburg writes about boys, girls and hot hands, comparing the problem with the Gilovich, Tversky, and Vallone paper to a probability question about the ratio of boys to girls in a village where families stop having children when they get a boy. I’m not sure I quite agree with him that the situations are analogous, but he has a good explanation of the hot hand issue, and there are some interesting threads in the comments.

It seems that these probability discussions tend to spill into the dreaded questions about boys born on Tuesdays. I have always bristled at these types of puzzles because they feel like semantics exercises rather than probability exercises. Once again, Jordan Ellenberg writes about them for Slate, leading off with this hilarious exchange:

Toby and Marla are playing a game with coins, because that’s what people in math problems do. Toby flips a fair coin three times, out of Marla’s view. “Did you get any heads?” Marla asks
“Yes,” Toby says. “For instance, the second coin came up heads.” (Because that’s how people in math problems talk.)
“I’ll bet you the next flip after that came up tails,” Marla says.
Is this a good bet?

Ellenberg’s point is that puzzles like this are psychological to some extent rather than mere exercises in probability—the right answer depends on our assumptions about how people communicate about coin flips or children and when they were born (mod 7). He also points to a great post by Tanya Khovanova about sons born on Tuesday which she followed up with an invented dialogue about the problem. I think I will refer to her posts the next time someone tries to trick me with a riddle from that genre.

Do you have any sons born on Tuesday? If so, what is the probability that you also have a daughter? Does she tend to make more baskets after making a few in a row, or does her basketball shooting record support the null hypothesis?

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