Mathematical proof for hot hand shooting in basketball

Basketball fans, players and coaches all swear by the "hot hand," the idea that players can experience winning streaks, being more likely to make a shot after having just made their last two or three shots. However, for decades, researchers repeatedly disproved the existence of the hot hand, claiming it was little more than a cognitive illusion. But this summer, in the latest twist to the story, economists Adam Sanjurjo and Joshua Miller of the universities of Alicante (UA) and Bocconi, respectively, published a working paper suggesting that this prior body of research is flawed by selection bias and that the hot hand might not be a fallacy after all.

The researchers began their paper with a basic example to illustrate their findings: an experiment in which a person throws a coin in the air four times. "We noted the results of each throw, and calculated the percentage of heads tossed immediately after another head for all possible throw sequences. The results were unexpected: the proportion is not 50%, as we intuitively believe, but actually around 40%," Sanjurjo tells us, adding that "these results suggest the existence of a bias."

Extrapolating to the world of basketball and the hot hand, this means that if a player's overall field goal percentage is 50%, and the average goal percentage for shots taken following a streak of three shots is also 50%, although counter-intuitive, statistically this validates the phenomenon of hot hand shooting.

Specifically, there results show that the average difference in goal percentage when on a streak is +13 percentage points. This becomes particularly significant when you consider that the difference between the best three-point shooter in the 2013-2014 NBA league and the median shooter was +10 percentage points.

Both economists have published two further articles on winning streaks over multiple shot databases. "We have found a strong evidence of hot hand shooting under various conditions, among players ranging from university students to semi-professionals," Sanjurjo said.