1.7 Why study coins, dice, cards, and spinners?

Many probability problems involve “toy” situations like flipping coins, rolling dice, shuffling cards, or spinning spinners. These situations might seem unexciting, or at least not very practically meaningful. However, coins and spinners and the like provide familiar, concrete situations which facilitate understanding of probability concepts. Furthermore, simple situations often provide insight into real and complex problems. The following is just one illustration.

Many basketball players and fans alike believe in the “hot hand” phenomenon: the idea that making several shots in a row increases a player’s chances of making the next shot. However, the consensus conclusion of thirty years of studies on the hot hand, beginning with the seminal study Gilovich, Vallone, and Tversky (1985), had been that there is no statistical evidence that the hot hand in basketball is real. As a result, many statisticians regularly caution against the “hot hand fallacy”: the belief that the hot hand exists when, in reality, the degree of streaky behavior typically observed in sequential data is consistent with what would be expected simply by chance in independent trials.

The idea behind studies like Gilovich, Vallone, and Tversky (1985) is essentially the following. Consider a player who attempts 100 shots and makes 50%. If there is no hot hand, then we might expect the player to make 50% of shots both on attempts that follow hit streaks — usually considered three (or more) made attempts in a row — and on other attempts. Therefore, a success rate of 50% on both sets of attempts provides no evidence of the hot hand.

However, recent research of Miller and Sanjurjo (2018a), Miller and Sanjurjo (2018c), Miller and Sanjurjo (2018b) concludes that previous studies on the hot hand in basketball, starting with Gilovich, Vallone, and Tversky (1985), have been subject to a bias. After correcting for the bias, the authors find strong evidence in favor of the hot hand effect in basketball shooting, suggesting the hot hand fallacy is not a fallacy after all. One interesting aspect of these studies is that Miller and Sanjurjo’s methods are simulation-based.

Miller and Sanjurjo (2018a) introduced the coin flipping problem in Section 1.6 to illustrate the idea behind their research and the bias in previous studies. Consider again a player who attempts 100 shots and makes 50%. Even if there is no hot hand, Miller and Sanjurjo show that we would actually expect the player to have a shooting percentage of strictly less than 50% on the attempts which followed streaks, and strictly greater than 50% on the other attempts. The reason is the same as for the coin flipping problem in Section 1.6: in a fixed number of trials, the proportion of H on trials following H is expected to be less than the true probability of H, even though the trials are independent. Therefore, for the example player a success rate of 50% on both sets of attempts actually provides directional evidence in favor of the hot hand. Properly acccounting for this bias leads to substantially different statistical analyses (i.e., p-values) and conclusions.


Gilovich, Thomas D., Robert P. Vallone, and Amos Tversky. 1985. “The Hot Hand in Basketball: On the Misperception of Random Sequences.” Cognitive Psychology 17 (3): 295–314. https://doi.org/https://doi.org/10.1016/0010-0285(85)90010-6.
Miller, Joshua B., and Adam Sanjurjo. 2018a. “Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers.” Econometrica 86 (6): 2019–47. https://doi.org/10.3982/ECTA14943.
———. 2018b. “A Cold Shower for the Hot Hand Fallacy: Robust Evidence That Belief in the Hot Hand Is Justified.” OSF Preprints. https://doi.org/10.31219/osf.io/pj79r.
———. 2018c. “Is It a Fallacy to Believe in the Hot Hand in the NBA Three-Point Contest?” OSF Preprints. https://doi.org/10.31219/osf.io/dmksp.