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COGNITIVE PSYCHOLOGY 17, 295-314 (1985)

The Hot Hand in Basketball: On the Misperceptionof Random SequencesTHOMAS GILOVICH

Cornell UniversiiyAND

ROBERT~ALLONE ANDAMOSTVERSKYStanford Uni\sersity

We investigate the origin and the val idity of common beliefs regarding the hothand and streak shooting in the game of basketball. Basketball players andfans alike tend to believe that a players chance of hitting a shot are greaterfollowing a hit than following a miss on the previous shot. However, detailedanalyses of the shooting records of the Philadelphia 76ers provided no evidencefor a positive correlation between the outcomes of successive shots. The sameconclusions emerged from free-throw records of the Boston Celtics, and from acontrolled shooting experiment with the men and women of Cornells varsityteams. The outcomes of previous shots influenced Cornell players predictionsbut not their performance. The belief in the hot hand and the detection ofstreaks in random sequences is attributed to a general misconception of chanceaccording to which even short random sequences are thought to be highly rep-resentative of their generating process. G 1985 Academic Press. In c.

In describing an outstanding performance by a basketball player, re-porters and spectators commonly use expressions such as Larry Birdhas the hot hand or Andrew Toney is a streak shooter. These phrasesexpress a belief that the performance of a player during a particular periodThis research was supported in part by a faculty research grant from the College of Artsand Sciences at Cornell University to the first author and by Grant NR 197-058 from theU.S. Off ice of Naval Research to the third author. We thank Harvey Pollack of the Phila-delphia 76ers and Todd Rosensweig of the Boston Celtics for providing their teams shootingstatistics, to Billy Cunningham of the 76ers for allowing his team to be interviewed, and toCornells Tom Miller and Linda Lerch for recruiting their players for our shooting experi-ment. We also thank Kathy Stratton for collecting a large part of the data reported here.This work has benefited from discussions with Persi Diaconis, David Freedman, Lee Ross,and Brian Wandell. Send requests for reprints to Dr. Thomas Gilovich, Psychology De-partment, Cornell University, Ithaca, NY 14853.

295 OOIO-0285/85 7.50Copyright C 1985 by Academic Press. Inc.All rights of reproduction in any for m reserved.

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296 GILOVICH, VALLONE, AND TVERSKYis significantly better than expected on the basis of the players overallrecord. The belief in the hot hand and in streak shooting is sharedby basketball players, coaches, and fans, and it appears to affect theselection of plays and the choice of players. In this paper we investigatethe origin and the validity of these beliefs.Peoples intuitive conceptions of randomness depart systematicallyfrom the laws of chance. It appears that people expect the essentialcharacteristics of a chance process to be represented not only globally inthe entire sequence, but also locally, in each of its parts. For instance,people expect even short seqeunces of heads and tails to reflect the fair-ness of a coin and contain roughly 50% heads and 50% tails. This con-ception of chance has been described as a belief in the law of smallnumbers according to which the law of large numbers applies to smallsamples as well (Tversky & Kahneman, 1971). A locally representativesequence, however, deviates systematically from chance expectation: Itcontains too many alternations and not enough long runs.A conception of chance based on representativeness, therefore, pro-duces two related biases. First, it induces a belief that the probability ofheads is greater after a long sequence of tails than after a long sequenceof heads-this is the notorious gamblers fallacy (see, e.g., Tversky &Kahneman, 1974). Second, it leads people to reject the randomness ofsequences that contain the expected number of runs because even theoccurrence of, say, four heads in a row-which is quite likely in a se-quence of 20 tosses-makes the sequence appear nonrepresentative(Falk, 1981; Wagenaar, 1972).Sequences of hits and misses in a basketball game offer an interestingcontext for investigating the perception of randomness outside the psy-chological laboratory. Consider a professional basketball player whomakes 50% of his shots. This player will occasionally hit four or moreshots in a row. Such runs can be properly called streak shooting, how-ever, only if their length or frequency exceeds what is expected on thebasis of chance alone. The players performance, then, can be comparedto a sequence of hits and misses generated by tossing a coin. A playerwho produces longer sequences of hits than those produced by tossing acoin can be said to have a hot hand or be described as a streakshooter. Similarly, these terms can be applied to a player who has abetter chance of hitting a basket after one or more successful shots thanafter one or more misses.

This analysis does not attempt to capture all that poeple might mean Feller (1968) describes some striking examples of the nonintuitive character of chanceprocesses (e.g., matching birthdates or the change of sign in a random walk), which heattributes to faulty intuitions about chance and common misconceptions of the law ofaverages.

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THE HOT HAND IN BASKE TBALL 297by the hot hand or streak shooting. Nevertheless, we argue thatthe common use of these notions-however vague or complex-impliesthat players performance records should differ from sequences of headsand tails produced by coin tossing in two essential respects. First, theseterms imply that the probability of a hit should be greater following a hitthan following a miss (i.e., positive association). Second, they imply thatthe number of streaks of successive hits or misses should exceed thenumber produced by a chance process with a constant hit rate (i.e., non-stationarity).It may seem unreasonable to compare basketball shooting to cointossing because a players chances of hitting a basket are not the sameon every shot. Lay-ups are easier than 3-point field goals and slam dunkshave a higher hit rate than turnaround jumpers. Nevertheless, the simplebinomial model is equivalent to a more complicated process with thefollowing characteristics: Each player has an ensemble of shots that varyin difficulty (depending, for example, on the distance from the basket andon defensive pressure), and each shot is randomly selected from thisensemble. This process provides a more compelling account of the per-formance of a basketball player, although it produces a shooting recordthat is indistinguishable from that produced by a simple binomial modelin which the probability of a hit is the same on every trial.We begin with a survey that explores the beliefs of basketball fansregarding streak shooting and related phenomena. We then turn to ananalysis of field goal and free-throw data from the NBA. Finally, wereport a controlled experiment performed by the men and women ofCornells varsity teams that investigates players ability to predict theirperformance.

STUDY 1: SURVEY OF BASKETBALL FANSOne hundred basketball fans were recruited from the student bodies ofCornell and Stanford University. All participants play basketball at leastoccasionally (65% play regularly). They all watch at least 5 gamesper year (73% watch over 15 games per year). The sample included 50captains of intramural basketball teams.The questionnaire examined basketball fans beliefs regarding sequen-tial dependence among shots. Their responses revealed considerableagreement: 91% of the fans believed that a player has a better chanceof making a shot after having just made his last two or three shots than

he does after having just missed his last two or three shots; 68% of thefans expressed essentially the same belief for free throws, claiming thata player has a better chance of making his second shot after making hisfirst shot than after missing his first shot; 96% of the fans thought thatafter having made a series of shots in a row . . . players tend to take

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298 GILOVICH, VALLONE, AND TVERS KYmore shots than they normally would; 84% of the fans believed that itis important to pass the ball to someone who has just made several (two,three, or four) shots in a row.The belief in a positive dependence between successive shots was re-flected in numerical estimates as well. The fans were aked to consider ahypothetical player who shoots 50% from the field. Their average esti-mate of his field goal percentage was 61% after having just made ashot, and 42% after having just missed a shot. Moreover, the formerestimate was greater than or equal to the latter for every respondent.When asked to consider a hypothetical player who shoots 70% from thefree-throw line, the average estimate of his free-throw percentage was74% for second free throws after having made the first, and 66% forsecond free throws after having missed the first.Thus, our survey revealed that basketball fans believe in streakshooting. It remains to be seen whether basketball players actuallyshoot in streaks.

STUDY 2: PROFESSIONAL BASKETBALL FIELD GOAL DATAField goal records of individual players were obtained for 48 homegames of the Philadelphia 76ers and their opponents during the 1980-

1981 season. These data were recorded by the teams statistician. Rec-ords of consecutive shots for individual players were not available forother teams in the NBA. Our analysis of these data divides into threeparts. First we examine the probability of a hit conditioned on playersrecent histories of hits and misses, second we investigate the frequencyof different sequences of hits and misses in players shooting records,and third we analyze the stability of players performance records acrossgames.Analysis of Conditional ProbabilitiesDo players hit a higher percentage of their shots after having just madetheir last shot (or last several shots), than after having just missed theirlast shot (or last several shots)? Table 1 displays these conditional prob-abilities for the nine major players of the Philadelphia 76ers during the1980- 1981 season. Column 5 presents the overall shooting percentage foreach player ranging from 46% for Hollins and Toney to 62% for Dawkins.Columns 6 through 8 present the players shooting percentages condi-tioned on having hit their last shot, their last two shots, and their lastthree shots, respectively. Columns 2 through 4 present the playersshooting percentages conditioned on having missed their last shot, theirlast two shots, and their last three shots, respectively. Column 9 presentsthe (serial) correlation between the outcomes of successive shots.A comparison of columns 4 and 6 indicates that for eight of the nine

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