The Cold Facts About the "Hot Hand" in Basketball Do basketball players tend to shoot in streaks? Contrary to the belief of fans and commentators, analysis shows that the chances of hitting a shot are as good after a miss as after a hit. Amos Tversky and Thomas Gilovich You're in a world all your own. It's sample, but also locally in each of its probability of alternation, p(a), or the hard to describe. But the basket seems parts. For example, people expect even probability that the outcome of a given to be so wide. No matter what you short sequences of heads and tails to shot would be different from the outcome do, you know the ball is going to go reflect the fairness of a coin and to of the previous shot. In a random (i.e., in. contain roughly 50% heads and 50% independent) sequence, p(a) = .5; streak –Purvis Short, of the NBA's Golden tails. Such a locally representative shooting and alternating shooting arise State Warriors sequence, however, contains too many when p(a) is less than or greater than .5, alternations and not enough long runs. respectively. Each respondent evaluated six sequences, with p(a) ranging from .4 This statement describes a phenomenon This misconception produces two to .9. Two (mirror image) sequences were known to everyone who plays or watches systematic errors. First, it leads many used for each level of p(a) and presented the game of basketball, a phenomenon people to believe that the probability of to different respondents. known as the "hot hand." The term refers heads is greater after a long sequence of to the putative tendency for success (and tails than after a long sequence of heads; The percentage of respondents who failure) in basketball to be self-promoting this is the notorious gamblers' fallacy. classified each sequence as "streak or self-sustaining. After making a couple Second, it leads people to question the shooting" or "chance shooting" is of shots, players are thought to become randomness of sequences that contain the presented in Figure 1 as a function of relaxed, to feel confident, and to "get in a expected number of runs because even the p(a). (The percentage of "alternating groove" such that subsequent success occurrence of, say, four heads in a row– shooting" is the complement of these becomes more likely. The belief in the which is quite likely in even relatively values.) As expected, people perceive hot hand, then, is really one version of a small samples– makes the sequence streak shooting where it does not exist. wider conviction that "success breeds appear non representative. Random The sequence of p(a) = .5, representing a success" and "failure breeds failure" in sequences just do not look random. perfectly random sequence, was classified many walks of life. In certain domains it as streak shooting by 65% of the surely does–particularly those in which a Perhaps, then, the belief in the hot respondents. Moreover, the perception of person's reputation can play a decisive hand is merely one manifestation of this chance shooting was strongly biased role. However, there are other areas, such fundamental misconception of the laws of against long runs: The sequences selected as most gambling games, in which the chance. Maybe the streaks of consecutive as the best examples of chance shooting belief can be just as strongly held, but hits that lead players and fans to believe were those with probabilities of where the phenomenon clearly does not in the hot hand do not exceed, in length alternation of .7 and .8 instead of .5. exist. or frequency, those expected in any random sequence. It is clear, then, that a common What about the game of basketball? misconception about the laws of chance Does success in this sport tend to be To examine this possibility, we first can distort people's observations of the self-promoting? Do players occasionally asked a group of 100 knowledgeable game of basketball: Basketball fans get a "hot hand"? basketball fans to classify sequences of detect" evidence of the hot hand in 21 hits and misses (supposedly taken perfectly random sequences. But is this Misconceptions of Chance from a basketball player's performance the main determinant of the widespread Processes record) as streak shooting, chance conviction that basketball players shoot shooting, or alternating shooting. Chance in streaks? The answer to this question One reason for questioning the widespread shooting was defined as runs of hits and requires an analysis of shooting statistics belief in the hot hand comes from misses that are just like those generated in real basketball games. research indicating that people's intuitive by coin tossing. Streak shooting and conceptions of randomness do not alternating shooting were defined as runs Cold Facts from the NBA conform to the laws of chance. People of hits and misses that are longer or commonly believe that the essential shorter, respectively, than those observed Although the precise meaning of terms characteristics of a chance process are in coin tossing. All sequences contained like "the hot hand" and streak shooting" represented not only globally in a large 11 hits and 10 misses, but differed in the is unclear, their common use implies a from pp16-21 of CHANCE: VOL 2, No. 1, 1989 1 shooting record that departs from coin "hot" stretches embedded in longer shooting. Nevertheless, it is instructive tossing in two essential respects (see stretches of normal performance. To to examine the performance of players accompanying box). First, the frequency obtain a more sensitive test of when the difficulty of the shot and the of streaks (i.e., moderate or long runs of stationarity (suggested by David defensive pressure are held constant. Free- successive hits) must exceed what is Freedman) we partitioned the entire record throw records provide such data. Free expected by a chance process with a of each player into non-overlapping series throws are shot, usually in pairs, from constant hit rate. Second, the probability of four consecutive shots. We then the same location and without defensive of a hit should be greater following a hit counted the number of series in which the pressure. If players shoot in streaks, their than following a miss, yielding a positive player's performance was high (3 or 4 shooting percentage on the second free serial correlation between the outcomes of hits), moderate (2 hits) or low (0 or 1 throws should be higher after having successive shots. hits). If a player is occasionally 'hot," his made their first shot than after having record must include more missed their first shot. Table 2 presents To examine whether these patterns high-performance series than expected by the probability of hitting the second free accurately describe the performance of chance. The numbers of high, moderate, throw conditioned on the outcome of the players in the NBA, the field-goal records and low series for each of the nine first free throw for nine Boston Celtics of individual players were obtained for 48 Philadelphia 76ers were compared to the players during the 1980-81 and the home games of the Philadelphia 76ers expected values, assuming independent 1981-82 seasons. during the 1980-81 season. Table 1 shots with a constant hit rate (taken from presents, for the nine major players of the column 5 of Table 1). For example, the These data provide no evidence that the 76ers, the probability of a hit conditioned expected percentages of high-, moderate-, outcome of the second shot depends on on 1, 2, and 3 hits and misses. The and low performance series for a player the outcome of the first. The correlation overall hit rate for each player, and the with a hit rate of .50 are 31.25%, 37.5%, is negative for five players and positive number of shots he took, are presented in and 31.25%, respectively. The results for the remaining four, and in no case column 5. A comparison of columns 4 provided no evidence for non-stationarity does it approach statistical significance. and 6 indicates that for eight of the nine or streak shooting as none of the nine players the probability of a hit is actually chi-squares approached statistical higher following a miss (mean = .54) significance. The analysis was repeated The Cold Facts from Controlled than following a hit (mean = .51), four times (starting the partition into Experiments contrary to the stated beliefs of both quadruples at the first, second, third, and players and fans. Column 9 presents the fourth shot of each player), but the To test the hot hand hypothesis, under (serial) correlations between the outcomes results were the same. Combining the controlled conditions, we recruited 14 of successive shots. These correlations four analyses. the overall observed members of the men's varsity team and are not significantly different than zero percentages of high, medium, and low 12 members of the women's varsity team except for one player (Dawkins) whose series are 33.5%, 39.4%, and 27.1%, at Cornell University to participate in a correlation is negative. Comparisons of respectively, whereas the expected shooting experiment. For each player, we the other matching columns (7 vs. 3, and percentages are 34.4%, 36.8%, and determined a distance from which his or 8 vs. 2) provide further evidence against 28.8%. The aggregate data yield slightly her shooting percentage was roughly streak shooting. Additional analyses fewer high and low series than expected 50%, and we drew two 15-foot arcs at show that the probability of a hit (mean by independence, which is the exact this distance from which the player took = .57) following a "cold" period (0 or 1 opposite of the pattern implied by the 100 shots, 50 from each arc.