The Hot Hand and The Cold Hand

Early in the afternoon on April 13, 2019, something surprising happened… Chris Davis got a hit.  Normally, there would not be much attention paid to a 1st Inning single by Davis, a baseball player for the Baltimore Orioles.  But, as Davis stood on 1st Base, amid cheers from the opposing fans in Boston’s Fenway Park, it was big news in the baseball world.

Why was a simple hit by Davis, a $17M/year star, so surprising?  Because, until that game, he had been hitless in his first 33 at-bats of the season, and in 54 consecutive at-bats overall, stretching from the end of the prior season — the longest hitless streak by a non-pitcher in the history of Major League Baseball.  From a straight-up statistical standpoint, using Davis’s .237 lifetime batting average, the probability of Davis going hitless in 54 consecutive at bats was 0.763 to the 54th power, or 0.763^54 = 0.000000453 — that is, there was a less than 1 out of 2 million chance that Davis would go hitless in those 54 consecutive at-bats.

Then, perhaps more surprising, in Davis’s next 4 at bats in the game, he had two more hits, both doubles.  And in his 12 games played following the streak (stretching to the time of this writing), he has gone a total of 12 for 35 (a 34.3% success rate) — using the same .237 lifetime batting average and binomial probability, there was only a 10% chance of him getting at least 12 hits in 35 at-bats… not crazy low, but given the futility streak preceding the stretch, it seems pretty unexpected.

Or does it?

WHAT IS A "HOT HAND”?

A “Hot Hand” is a general catch-all term referring to the idea that a successful attempt in a probabilistic effort means the next instance will also have a higher likelihood of success.  That is, if a basketball player just hit a couple of shots in a row, he’s more likely to make the next attempt than if he had missed those prior shots. Or that if the Super Bowl coin flip has come up tails 4 years in a row, it’s more likely to come up heads this year.

The issue that mathematicians have pointed out with this “hot hand” idea — attributing success leading to success — is that, from a statistical standpoint, we should see streaks of successes by pure chance.

For example, there’s a 1/6 chance of rolling a 7 with two dice, so there’s a (roughly) 7,500 to 1 chance that 5 consecutive rolls would all be 7s.  Now, those odds may seem pretty low but given all the rolling that people do, you may at some point see someone roll 5 consecutive 7s… and does that make the probability the next roll will be a 7 greater than 1 out of 6?  Well, unless you’re playing with loaded dice, the answer would be “obviously not” — but if you're watching someone roll all those 7s, you may expect the next roll to be a 7 as well…because the roller may just have a “hot hand”.

Or consider Klay Thompson, a superstar for basketball’s Golden State Warriors.  From a pure probabilistic standpoint, given Klay's lifetime 40.2% 3-point shot percentage, there’s a 1 out of 3,647 chance that he would hit 9 consecutive 3-pointers.  If he does, is there a greater than 40.2% chance that he’ll hit his next one? Isn’t this situation different than the dice roller, since (defense readjustments aside) Klay actually can affect the result of his shot?

WHAT IS THE "HOT HAND FALLACY”?

Well, according to (many) mathematicians, the answer is “no”, the dice and Klay situations are the same.  The theory is that, even though we have the illusion that Klay's ability to affect his own shot means that an observed success in his last shot is telling us something about his next shot, this influence is already baked into his shooting percentage.  From this view, Klay's statistical likelihood of success in any particular shot is a pre-determined probability in the same way that a dice roll's pre-determined probability of being a 7 is always 1/6.

Because this position denies the idea of a “Hot Hand,” it has become referred to as the “Hot Hand Fallacy”.  The “Hot Hand Fallacy” says that, from a purely statistical standpoint, the existence of “streaks” (e.g. consecutive free throws made, etc.) is completely explainable as falling within reasonable bounds of the normal distribution based on a player’s overall average.

WHAT IS THE "HOT HAND FALLACY” FALLACY?

The problem with the “Hot Hand Fallacy” is that athletes are not robots — programmed to succeed at a certain average rate every game, so their performance is just based on the resulting probabilistic distribution.  Rather, they are humans, who — despite all the muscle memory and automaticity of a shot, or a swing, or a throw — are affected by both emotion and current physical condition.

When Chris Davis was already midway through his hitless streak, he wasn’t the same .237 hitter with a 23.7% chance of getting a hit in his next at-bat.  Rather, he was thinking about his streak, making him both mentally and physically tighter — and that negatively affected his chance of success in each next at-bat.  Only when he got that hit to end his streak, was he able to restore his confidence, and begin to even outperform his career average. We could call the hitless streak a “Cold Hand” rather than a “Hot Hand” but it is exactly analogous.

Klay Thompson did make 9 consecutive 3-pointers in a game back in January.  Then he hit his 10th as well. The chances of Klay hitting 10 in a row was 9,073 to 1.  And while he missed his 11th, it still made his 3-point shooting on the day 10 for 11 (Hot Hand!) — only one week after a Jan 13 game in which he went 2 for 11 (Cold Hand!).  In which game do you think Klay was feeling more confident in his shot — and thereby affecting its probability of success?

That is not something that happens with dice-rolls.

RESOLVING THE "HOT HAND FALLACY” FALLACY

Mathematicians are not wrong that streaks could be explainable based on overall average.  But a better way of thinking about athletic performance is that a player’s overall average is not a static number.  Every game a player comes in as a slightly different version of himself or herself… and even in-game performance can influence a player’s confidence, and resulting performance, for the rest of the game.  Knowing you hit your first few 3-pointers does make it a little more likely you’ll make your next one. And knowing that you haven’t gotten a hit in, say, 40 straight at bats, makes you press at the plate, lowering your chances of success in your next at-bat.

Combining all these game-by-game performances does create an overall average — where any of the per-game performances are statistically plausible.  But Chris Davis after that streak-ending hit on April 13 was not the same hitter as he was in the prior 54 at-bats.

And if I’m on the Warriors… if I see Klay hit his first couple of 3-pointers, I’m looking to get him the ball.

Thanks for reading! Feel free to email me your thoughts.