Hot Hands of Randomness

A lot of basketball fans believe in hot hands and streak shooting. These phrases roughly suggest the probability of a basket is higher following a hit than a miss. A spectator suddenly gets confidence in a player who shot the previous two shots to make the next. At that moment, she would not think about probabilities (coin tossing games) but only consider the momentum and confidence of the shooter.

It is interesting to notice that this behaviour of the fans is just the opposite of the belief in “the law of small numbers” or the gambler’s fallacy. The reason: basketball is being played by individuals with flesh and blood (and lucky charm), whereas gambling is by machines that work on chances (and expected values).

In 1985, Gilovich et al. did a study that analysed the data from 48 home games of the Philadelphia 76ers and their opponents during the 1980-81 season. The following is what they found.

The first table is about the (conditional) probability of a hit, given the player had already hits (3, 2 and 1). P(hit) represents the overall shooting percentage of the player.

	P(hit)	P(hit|3H)	P(hit|2H)	P(hit|1H)
Richardson	0.50	0.48	0.50	0.49
Erving	0.52	0.48	0.52	0.53
Hollins	0.46	0.32	0.46	0.46
Cheeks	0.56	0.59	0.54	0.55
Jones	0.47	0.27	0.43	0.45
Toney	0.46	0.34	0.40	0.43
Jones	0.54	0.53	0.47	0.53
Mix	0.52	0.36	0.48	0.51
Dawkins	0.62	0.51	0.58	0.57

Similar statistics for a player to hit after missing the previous shots are presented below.

	P(hit)	P(hit|3M)	P(hit|2M)	P(hit|1M)
Richardson	0.50	0.50	0.47	0.56
Erving	0.52	0.52	0.51	0.51
Hollins	0.46	0.50	0.49	0.46
Cheeks	0.56	0.77	0.60	0.60
Jones	0.47	0.50	0.48	0.47
Toney	0.46	0.52	0.53	0.51
Jones	0.54	0.61	0.58	0.58
Mix	0.52	0.70	0.56	0.52
Dawkins	0.62	0.88	0.73	0.71

The data showed just the opposite of what people believed. The hits following previous hits were lower than previous misses.

Gilovich, T.; Vallone, R.; Tversky. A., Cognitive Psychology 17, 295-314 (1985)