Biases in casino betting: The hot hand and the gambler's fallacy

Judgment and Decision Making, Vol. 1, No. 1, July 2006, pp. 1–12
Biases in casino betting: The hot hand and the gambler’s fallacy
James Sundali?
Rachel Croson
Managerial Sciences
Operations and Information Management
University of Nevada, Reno
Wharton School
University of Pennsylvania
Abstract
We examine two departures of individual perceptions of randomness from probability theory: the hot hand and the
gambler’s fallacy, and their respective opposites. This paper’s ?rst contribution is to use data from the ?eld (individuals
playing roulette in a casino) to demonstrate the existence and impact of these biases that have been previously docu-
mented in the lab. Decisions in the ?eld are consistent with biased beliefs, although we observe signi?cant individual
heterogeneity in the population. A second contribution is to separately identify these biases within a given individual,
then to examine their within-person correlation. We ?nd a positive and signi?cant correlation across individuals between
hot hand and gambler’s fallacy biases, suggesting a common (root) cause of the two related errors. We speculate as to
the source of this correlation (locus of control), and suggest future research which could test this speculation.
Keywords: judgment and decision making, hot hand, gambler’s fallacy, casino betting, ?eld data, roulette
1 Introduction
relation of a non-autocorrelated random sequence of out-
comes like coin ?ips. For example, imagine Jim repeat-
Almost every decision we make involves uncertainty in
edly ?ipping a (fair) coin and guessing the outcome be-
some way. Yet research on decision making under un-
fore it lands. If he believes in the gambler’s fallacy, then
certainty demonstrates that our judgments are often not
after observing three heads in a row, his subjective prob-
consistent with probability theory. Intuitive ideas of ran-
ability of seeing another head is less than 50%. Thus he
domness depart systematically from the laws of chance.
believes a tail is “due,” and is more likely to appear on
This research suggests that we have developed a number
the next ?ip than a head.
of judgment heuristics for analyzing complex, real-world
In contrast, the hot hand is a belief in positive auto-
events. Although many decisions based on these heuris-
correlation of a non-autocorrelated random sequence of
tics are consistent with probability theory, there are also
outcomes like winning or losing. For example, imagine
situations where heuristics lead to statistical illusions and
Rachel repeatedly ?ipping a (fair) coin and guessing the
suboptimal actions.
outcome before it lands. If she believes in the hot hand,
This paper investigates the existence and impact of two
then after observing three correct guesses in a row her
of these statistical illusions; the gambler’s fallacy and
subjective probability of guessing correctly on the next
the hot hand. Both of these illusions characterize in-
?ip is higher than 50%. Thus she believes that she is
dividuals’ perceptions of non-autocorrelated random se-
“hot” and more likely than chance to guess correctly.
quences. Thus both involve perceptions of sequences of
events rather than one-time events.
Notice that these two biases are not simply opposites.
The gambler’s fallacy is a belief in negative autocor-
The gambler’s fallacy describes beliefs about outcomes
of the random process (e.g., heads or tails), while the hot
?The authors thank Eric Gold for substantial contributions in ear-
hand describes beliefs of outcomes of the individual (like
lier stages of this research. Thanks also to Jeremy Bagai, Dr. Klaus
von Colorist, Bradley Ruf?e, Paul Slovic, Willem Wagenaar, partici-
wins and losses). In the gambler’s fallacy, the coin is due;
pants of the J/DM and ESA conferences, at the Conference on Gam-
in the hot hand the person is hot. For purposes of our
bling and Risk Taking and at seminars at Wharton, Caltech and IN-
study, we will identify four possible biases that individu-
SEAD for their comments on this paper. Special thanks to the In-
als could exhibit. The gambler’s fallacy and its opposite,
stitute for the Study of Gambling and Commercial Gaming for in-
dustry contacts which resulted in the acquisition of the observational
the hot outcome, are beliefs about the coin’s outcomes
data reported here. Financial support from NSF SES 98–76079–001
involving negative versus positive autocorrelation of ran-
is also gratefully acknowledged. All remaining errors are ours. Ad-
dom outcomes. The hot hand and its opposite, the stock
dress correspondence to Rachel Croson, 567 JMHH, The Wharton
of luck, are beliefs about the individual’s success involv-
School University of Pennsylvania, 3730 Walnut Street, Philadelphia,
PA 19104–6340, crosonr@wharton.upenn.edu. James Sundali’s email
ing positive versus negative autocorrelation of winning or
is jsundali@unr.nevada.edu
losing.
1

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Judgment and Decision Making, Vol. 1, No. 1, July 2006
Biases in casino betting
2
Thus someone can believe both in the gambler’s fallacy
ture on probability matching. In these experiments sub-
(that after three coin ?ips of heads tails is due) and the
jects were asked to guess which of two colored lights
hot hand (that after three wins they will be more likely to
would next illuminate. After seeing a string of one out-
correctly guess the next outcome of the coin toss). These
come, subjects were signi?cantly more likely to guess
biases are believed to stem from the same source, the rep-
the other, an effect referred to in that literature as neg-
resentativeness heuristic, as discussed below (Gilovich,
ative recency (see Estes, 1964, and Lee, 1971, for re-
Vallone and Tversky 1985).
views). Ayton and Fischer (2004) also demonstrate the
In this paper we use empirical data from gamblers in
existence of gambler’s fallacy beliefs in the lab when sub-
casinos to examine the existence, prevalence and correla-
jects choose which of two colors will appear next on a
tion between gambler’s fallacy and hot hand beliefs. A
simulated roulette wheel. Gal and Baron (1996) show
companion paper, Croson and Sundali (2005) uses the
that gambler’s fallacy behavior is not simply caused by
same data to examine the aggregate (market) impact of
boredom; participants in their experiments were asked
these biases. In contrast, here we will identify the biases
how they would best maximize their earnings, and they
at the individual level, and examine the within-participant
responded with gambler’s fallacy type logic.
correlation between the two.
The gambler’s fallacy is thought to be caused by
Empirical data, while dif?cult to obtain and to code,
the representativeness heuristic (Tversky and Kahneman
can provide an important complement and robustness
1971, Kahneman and Tversky 1972). Here, chance is per-
check on other methods in investigating biases. Partici-
ceived as “a self-correcting process in which a deviation
pants in the casinos are making real decisions with their
in one direction induces a deviation in the opposite direc-
own money on the line. Further, the participants repre-
tion to restore the equilibrium” (Tversky & Kahneman,
sent a more motivated sample than typical students at a
1974, p. 1125). Thus after a sequence of three red num-
university; gamblers have a very real incentive to learn
bers appearing on the roulette wheel, black is more likely
the game they are playing and to make decisions in ac-
to occur than red because a sequence red-red-red-black is
cordance with their beliefs.
more representative of the underlying distribution than a
The use of casino data does, however, involve some
sequence red-red-red-red. We test for the gambler’s fal-
limitations. In particular, we were prevented from di-
lacy in our data by looking at the impact of previous out-
rectly contacting the gamblers in the study, thus we can-
comes on current bets at roulette. People who believe
not ask particular individuals why they bet how they did
in the gambler’s fallacy should be less likely to bet on a
or about their beliefs at the time of placing the bet. Also,
number that has previously appeared.
the gambling population, while motivated, is a selected
For purposes of this analysis, we will examine two sep-
subsample of the population at large. Thus we will have
arate de?nitions of hotness, hot outcome and hot hand.
to be cautious in our claims of external validity from this
Hot outcome will simply be the opposite of the gambler’s
study. Nonetheless, we believe that the demonstration of
fallacy, that is, an (incorrect) belief in positive autocor-
these biases in the ?eld at the level of the individual is an
relation of a non-autocorrelated random sequence.2 For
important contribution in and of itself. We are also one
example, individuals who believe in hot outcome believe
of the very few papers to identify multiple biases within
that after three red numbers appearing on the roulette
an individual and to characterize the correlation between
wheel, another red number is more likely to appear than
them.
a black number because red numbers are hot. Notice that
here the outcomes are hot (e.g., red numbers), rather than
1.1 De?nitions and previous research
individuals, as in the hot hand, below.
In the lab, the literature on probability matching also
1.1.1 Gambler’s fallacy
provides evidence favoring hot outcome beliefs. Ed-
wards (1961), Lindman and Edwards (1961) and Feld-
The gambler’s fallacy is de?ned as an (incorrect) belief in
man (1959) all found positive recency effects in proba-
negative autocorrelation of a non-autocorrelated random
bility matching tasks. In particularly long sequences of
sequence.1 For example, individuals who believe in the
the probability matching game, participants were signi?-
gambler’s fallacy believe that after three red numbers ap-
cantly more likely to guess the same outcome as had been
pearing on the roulette wheel, a black number is “due,”
observed previously.3
that is, is more likely to appear than a red number.
Gambler’s fallacy-type beliefs were ?rst observed in
2Or, more generally, a belief in a more positive autocorrelation than
the laboratory (under controlled conditions) in the litera-
is present. Thus when an individual overestimates the amount of posi-
tive autocorrelation in any sequence, we could say they were exhibiting
1Or, more generally, a belief in a more negative autocorrelation than
hot outcome beliefs as well.
is present. Thus when an individual overestimates the amount of nega-
3The hot outcome bias is related but not identical to the construct
tive autocorrelation in any sequence, we could say they were exhibiting
referred to by Keren and Lewis (1994) as the gambler’s fallacy type II.
gambler’s fallacy beliefs as well.
They present results of a questionnaire study in the lab demonstrating

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Judgment and Decision Making, Vol. 1, No. 1, July 2006
Biases in casino betting
3
We will test for hot outcome beliefs in our data by
— this is the notorious gambler’s fallacy (see,
looking at the impact of previous outcomes on current
e.g., Tversky and Kahneman, 1971). Second,
bets at roulette. If gamblers believe in hot outcomes, they
it leads people to reject the randomness of se-
should be more likely to bet on an outcome that has pre-
quences that contain the expected number of
viously been observed. Thus a positive relationship be-
runs because even the occurrence of, say, four
tween previously-observed outcomes and current bets is
heads in a row — which is quite likely in a se-
indicative of a belief in hot outcomes.4
quence of 20 tosses — makes the sequence ap-
pear nonrepresentative. (p. 296).
1.1.2 Hot hand
This second explanation is supported by data in which
Hot hand is different from hot outcome. Rather than be-
participants are asked to generate strings of random num-
lieving that a particular outcome is hot, individuals who
bers. The strings generated produced signi?cantly fewer
believe in the hot hand believe that a particular person is
runs of the same outcome than a truly random sequence
hot. For example, if an individual has won in the past,
would (see Wagenaar 1972 for a review, for an exception
whatever numbers they choose to bet on are likely to win
see Rapoport & Budesceu 1992).
in the future, not just the numbers they’ve won with pre-
We will test for hot hand beliefs in our data by looking
viously.
at how betting behavior changes in response to wins and
Gilovich, Vallone and Tversky (1985) demonstrated
losses. In particular, hot hand beliefs predict that after
that individuals believe in the hot hand in basketball
winning, individuals will increase the number of bets they
shooting, and that these beliefs are not correct (i.e., bas-
place and after losing, decrease them.
ketball shooters’ probability of success is indeed serially
Just as the gambler’s fallacy and the hot outcome are
uncorrelated). Other evidence from the lab shows that
opposing biases, the hot hand has an opposing bias, re-
subjects in a simulated blackjack game bet more after a
ferred to here as “stock of luck” beliefs. Individuals be-
series of wins than they do after a series of losses, both
lieve they have a stock or ?xed amount of luck and, once
when betting on their own play and on the play of others
it’s spent, their probability of winning decreases. Thus
(Chau & Phillips, 1995). Further evidence of the hot hand
after a string of wins, individuals are less likely to win
in a laboratory experiment comes from Ayton and Fis-
(rather than more likely as predicted by the hot hand) be-
cher (2004). Participants exhibit more con?dent in their
cause they have exhausted their stock of luck. The ef-
guesses of what color will next appear after a string of
fect has been demonstrated in the lab by Leopard (1978)
correct guesses than after a string of incorrect guesses.
who examines choice behavior in a series of gambles and
Explanations for the hot hand are numerous. It is
demonstrates that subjects take more risk after losing than
clearly related to the illusion of control (Langer, 1975),
after winning, suggesting that their bad luck is about to
where individuals believe they can control outcomes that
change or their good luck about to run out.5
are, in fact, random. Gilovich et al., (1985) suggest
Stock of luck beliefs predict that after winning, indi-
that the hot hand also arises out of the representativeness
viduals will decrease the number of bets they place and,
heuristic, just as the gambler’s fallacy. They write
after a loss, increase them. Thus a negative relationship
observed between current betting behavior and previous
A conception of chance based on representa-
wins/losses will provide evidence for this bias.
tiveness, therefore, produces two related bi-
ases. First, it induces a belief that the proba-
bility of heads is greater after a long sequence
1.2 Individual differences
of tails than after a long sequence of heads
A large literature identi?es individual differences in risk
that individuals underestimate the number of observations necessary to
attitudes (e.g., Weber et al.,1992; Blais & Weber, 2006;
detect biased roulette wheels. Thus after seeing even a small streak of
Harris et al., 2006). In addition, previous work has iden-
red numbers, gamblers might believe the wheel is biased and expect
ti?ed individual heterogeneity in biased beliefs about se-
more red numbers. The number of spins participants believe they need
to observe to detect a biased wheel, while signi?cantly smaller than the
quences of gambles. Friedland (1988) uses a personality
true number of spins necessary, as derived in Ethier (1982), is signi?-
inventory to categorize individuals into luck-oriented and
cantly larger than the number of spins any individual in our data set will
observe.
5As with the hot outcome above, there are alternative explanations
4One can construct other explanations for the behavior we here at-
for these behaviors as well. For example, wealth effects or house money
tribute to the hot outcome. For example, perhaps numbers that have
effects might cause an increase in betting after a win (hot hand) (Thaler
recently hit on the roulette wheel are more available to the gambler
& Johnson 1990). Prospect theory’s assumption of increased risk-
than other numbers. This availability may cause the gambler to bet on
seeking in losses might cause an increase in betting after a loss (stock
numbers that have recently hit. Unfortunately in our empirical data we
of luck). In the lab, these effects can be separated by eliciting beliefs
will not be able to distinguish between these alternative causes of this
directly as in Ayton and Fischer (2004). In our empirical data we will
behavior, although previous lab research can and has done so.
not be able to distinguish between these alternative explanations.

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Judgment and Decision Making, Vol. 1, No. 1, July 2006
Biases in casino betting
4
chance-oriented. In a questionnaire design, he ?nds gam-
or baccarat where cards are dealt without replacement.
blers’ fallacy behavior in luck-oriented individuals but no
Second, each player has his or her own colored chips,
such behavior, and in particular, no dependence of current
thus tracking an individual’s betting behavior is feasible.
bets on past outcomes, in chance-oriented individuals.
Finally, roulette is an extremely popular and accessible
In the ?eld, previous work has also found individ-
game which requires relatively little skill to play (unlike
ual heterogeneity in biased beliefs. Keren and Wage-
craps, for example, which is perceived as a game for ex-
naar (1985) examine blackjack play of 47 individuals
perts). Thus roulette is likely to suffer from less selection
who played at least 75 hands and changed their bets over
bias than craps, although we are already selecting partic-
time. Of these, 25 had relationships between previous
ipants from the casino gambling population, mentioned
outcomes and bet changes (thus, exhibiting a bias of some
above as an unavoidable selection bias.
sort). Fourteen of them increased their bets after they won
and decreased them after they lost (consistent with the hot
1.3.1 Roulette
hand), while 11 decreased their bets after winning and in-
creased them after losing (consistent with stock of luck).
Roulette involves a dealer (sometimes two), a wheel and
As in these studies, we will use our data to analyze indi-
a layout. The wheel is divided into 38 even sectors, num-
vidual differences in betting behavior.
bered 1-36, plus 0 and 00. Each space is red or black,
Only two previous papers examine ?eld behavior at
with the 0 and 00 colored green. The wheel is arranged
roulette. The ?rst is an observational sociological ?eld
as shown in Figure 1, such that red and black numbers
study by Oldman (1974) which informally reports both
alternate.
the gambler’s fallacy and the hot outcome. He writes
that “[t]he bet on a particular spin tends to be placed on
outcomes that are ‘due’ either because they have not oc-
curred for some time or because that is the way ‘things
are running.”’ (p. 418). The second source, Wagenaar
(1988, Ch. 4), discusses data from 29 roulette players in
a casino who stayed between 1 and 18 spins each. Of
the 11 players who varied their bets most, he ?nds after
a win 39% of bets involve increased risk (hot hand) and
61% involve decreased risk (stock of luck). After a loss,
43% of bets involve decreased risk (hot hand) and 57%
of bets do not (stock of luck). However, Wagenaar does
not present an analysis of how individuals differ on this
dimension.
While previous papers have investigated the gambler’s
fallacy and hot hand biases, our work makes two impor-
tant and original contributions. First, it provides a ?eld
Figure 1: The wheel
setting in which it is possible to investigate both biases
at once. These biases have been analyzed together only
Players arrive at the roulette table and offer the dealer
in the lab (Ayton and Fischer 2004). Second, our empiri-
money (either cash or casino chips). In exchange, they
cal data will allow us to identify individual differences in
are given special roulette chips for betting at this wheel.
these biases. We will be able to examine the correlation
These chips are not valid anywhere else in the casino, and
between these biases within the individual.6
each player at the table has a unique color of chips. Play-
ers bet by placing chips on a numbered layout, the wheel
1.3 Field data
is spun and a small white ball rolled around its edge. The
ball lands on a particular number in the wheel, which
In this study we use observational data from the ?eld;
is the winning number for that round, and is announced
individuals betting at roulette in a casino. Roulette is a
publicly by the dealer. Next, the dealer clears away all
useful game for a number of reasons. First, it is serially
losing bets, players who had bet on the winning num-
uncorrelated, unlike other casino games like blackjack
ber (in some con?guration) are paid in their own-colored
6Our companion paper, Croson and Sundali (2005) has examined
chips and a new round of betting begins.
thee data at the aggregate level. There we provide evidence that the
Figure 2 shows a typical layout, along with the types
wheel is unbiased, that gambler’s fallacy behavior is observed in outside
of bets that can be made. Unlike the wheel, the layout is
bets after long streaks (5 and 6 observations of the same type), and that,
in aggregate, individuals place more bets after they have won a previous
arranged in numerical order. Players can place their bets
bet than after they have lost one (or than on their ?rst spin).
on varying places on the layout. Bets of the type on the

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Judgment and Decision Making, Vol. 1, No. 1, July 2006
Biases in casino betting
5
number 30 are called “straight up” bets. These are bets on
3-day period in July of 1998.9 The videotapes provided
a single number. If the number comes up on the wheel,
an overhead view of the roulette area. The camera an-
this bet would pay the player 36 for 1 (35 to 1). That is,
gle was focused on the roulette layout to allow the coding
when 1 chip is bet, the dealer pays the player 35 chips
of bets placed and to protect player anonymity. Players
directly, and the chip that was bet is not removed from
were not directly visible, however individual bets could
the table. Bets of the type between the 8 and 11, “line
be tracked by the color of the chips being used. The
bets” are bets on two numbers. If either of the numbers
videotape was subtitled with a time counter. Note that
comes up, this bet pays the player 18 for 1. Players can
while many casinos employ electronic displays showing
also bet on combinations of 3 numbers (by the 13) which
previous outcomes of the wheel, this casino had no such
pay 12 for 1, combinations of 4 numbers (on the corner
displays at the time the data was collected.
of 17-18-20-21) which pay 9 for 1, or combinations of 6
A research assistant was employed to view and record
numbers (by the 22-25) which pay 6 for 1. Players can, of
player bet data from these videotapes. Players were iden-
course, bet on “outside” bets like red/black, even/odd and
ti?ed based upon the color of the chips being used to bet,
low/high. These bets will not be included in our analysis,
the player’s location at the table, and any distinct char-
as they are not bet often enough to allow identi?cation at
acteristics of the player’s hand or arm such as jewelry,
the individual level, but are discussed in our companion
clothing, tattoos, etc. Players who ran out of chips and
paper on aggregate behavior, Croson and Sundali (2005).
immediately bought more (of the same color) were coded
as the same player. Players who ran out of chips and did
not immediately buy more were coded as having left the
table. When money was again exchanged for chips of that
particular color, we assumed a new player had joined the
table.10
The videotape methodology made it possible to view
all of bets made by each player with a high degree of
accuracy. However, while we could observe if a player
Figure 2: The layout
bet on a particular number, given the angle of the cam-
era (from above), we could not observe how many chips
Notice that all these bets have the same expected value,
he or she bet on a particular number. Thus we simpli?ed
-5.26% on a double-zero wheel.7 Since the house advan-
the data recording to include simply a bet being placed,
tage on (almost) all bets at the wheel is the same, there
without mention of how much the bet was. In order to be
is no economic reason to bet one way or another (or for
consistent in not recording the amount bet, we coded bets
that matter, at all). In this paper, we will compare ac-
on multiple numbers (fractional bets like those in Figure
tual betting behavior we observe against a benchmark of
2) the same as we recorded bets on single numbers. For
random betting and search for systematic and signi?cant
example, a player could place a single “corner bet” on
deviations from that benchmark.
17, 18, 20, 21 by placing his chip at the intersection of
these numbers. We recorded this bet as a bet placed on
each of the four numbers. We limit our analysis in this
2 Method
paper to bets placed in the inside of the roulette layout,
thus we do not count bets placed on black/red, even/odd,
The data were gathered from a large casino in Reno,
high/low, 1st, 2nd or 3rd 12 or columns in our data; the
Nevada, and were also used in Croson and Sundali (2005)
interested reader can ?nd analysis of these outside bets in
to examine aggregate behavior.8 Casino executives sup-
aggregate in Croson and Sundali (2005). After the assis-
plied the researchers with security videotapes for 18
hours of play of a single roulette table. The videotapes
9The three time blocks were from 4:00 p.m. to 10:00 p.m., 8:00
consisted of three separate six-hour time blocks over a
p.m. to 2:00 a.m., and 10:00 p.m. to 4:00 a.m. These time blocks were
appropriate since the majority of gaming business is done in the evening
7This statement is not strictly true. One bet has a house advantage
hours.
of 7.89%. The bet involves placing a chip on the outside corner of the
10This coding has the potential to introduce two possible errors; two
layout between 0 and 1. The bet wins if 0, 00, 1, 2 or 3 appears, but
different people could be counted as the same person, or the same per-
pays only 6 for 1 (as though the bet were covering 6 numbers instead
son could be counted as two different people. We believe that the ?rst of
of 5). We observed only 75 instances of this bet being placed (out of
these errors is minimized; when chips were depleted and someone im-
22,527 bets). Only 11 different individuals placed this bet (out of 139
mediately purchased more, the coder could recognize from their hand
identi?able individuals in our data), and of them, only 6 placed this bet
characteristics if it was the same person. Additionally, this casino has
more than twice.
many roulette tables, it was rare that this table was full or that people
8At the time of data collection a casino in Washoe County, Nevada,
were waiting to buy in immediately after someone had gone bust. The
was classi?ed as “large” by the Nevada Gaming Control Board if total
second error may be somewhat more likely, here we rely on the obser-
(yearly) gaming revenues for the property exceeds $36 million.
vation that if an individual wants to rebuy, (s)he rarely waits to do so.

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Judgment and Decision Making, Vol. 1, No. 1, July 2006
Biases in casino betting
6
tant recorded all of the bets from the 18 hours of video-
If players bet according to the gambler’s fallacy, the
tape, one of the principal investigators performed an audit
probability of their betting on a given number should be
check to insure accuracy.
negatively related to its hotness measure; numbers which
have come up more frequently while they were at the ta-
ble are less likely to be bet on. In contrast, if players
3 Results
bet according to the hot outcome, the probability of their
betting on a given number should be positively related to
3.1 Descriptive statistics: The wheel and
its hotness measure. Notice that this hotness measure is
the bets
calculated separately for each individual in each period,
based on what they have observed up to the point of plac-
Nine hundred and four spins of the roulette wheel were
ing their bets.
captured in this data set (approximately 1 spin per
The second independent variable is an attempt to con-
minute). The expected frequency of a single number on a
trol for the baseline bets of individuals. Roulette players
perfectly fair roulette wheel is 1/38 or 2.6%. In this sam-
often bet the same numbers consistently and repeatedly;
ple the most frequent outcome was number 30 at 3.7%,
the bets don’t vary with past outcomes. Thus, we need
the least frequent outcome was number 26 at 1.7%. These
to control for these bets. Some players get lucky and
data provide no evidence that the wheel is biased.11 Ta-
hit those numbers (and others don’t), which could cause
ble 1 presents the outcomes and the bets placed during
the ?rst type of players to look as though they were bet-
our sample.
ting numbers which had come up before and the second,
If players bet randomly, we would expect them to bet
those which hadn’t. Instead, we want to look at devia-
on each number equally, thus 2.6% of the bets should fall
tions from betting patterns as numbers come up. Thus
on each number, independently of the history of numbers
in the model, we include F
which have appeared. This independence is what we will
it, the percentage of spins on
which the player has bet on number i previously to period
test in our analyses.
t.
We expect the coef?cient on this variable to be sig-
3.2 Gambler’s fallacy vs. hot outcome
ni?cant and positive (if players bet on a number previ-
ously, they are more likely to bet on it again). However,
We will use a general linear model to analyze the prob-
our main reason for including it is to control for under-
ability of a bet being placed on a number that has previ-
lying personal preferences over numbers that might bias
ously appeared, versus one which has not. Our dependent
our coef?cient of interest, the hotness measure. Thus a
variable, Pit is binary; if a bet was placed on number i on
signi?cant coef?cient on the hotness measure measures a
spin t, we record a success (1). If no bet was so placed,
deviation from the expected betting pattern of an individ-
we record a failure (0). Thus we will try to predict, on the
ual, given their bets up until now.
basis of previous outcomes, whether a player will bet on
a particular number.
The ?nal independent variable, Lit, controls for a be-
Independent variables include an intercept, a measure
havioral anomaly particular to roulette. When a bet wins,
of the hotness of a number, a control for the player’s “fa-
the dealer pays the winnings directly to the player, but
vorite” numbers and a control for leaving a bet on the
leaves the winning chip on the same spot on the table.
table. We measure a number’s hotness by calculating a
Many players are reluctant to move this winning chip,
measure of how often the number i has appeared while
claiming it is unlucky. If we were to count that unmoved
the player was at the table in the spins before spin t. In
chip as a bet, we would bias the results toward hot out-
particular, H
come, as players are often betting (by default) on num-
it is how many times number i has appeared
while the player was at the table before round t minus
bers that have won in the previous round.12 We control
the expected frequency of the number i appearing. This
for this behavior by including an independent variable
expected frequency is simply (1-(37/38)t-1) where (t-1) is
that equals one if an individual has bet on a number in
the number of trials observed by the player so far. This
the previous round and it has won, and a zero otherwise.
hotness measure thus calculates the actual frequency of
12
a number appearing minus the expected frequency. If a
This behavior is consistent with the status quo bias (Samuelson and
Zeckhauser 1988) or the omission/commission bias (Ritov and Baron
number has appeared more than expected, this hotness
1992, Baron and Ritov 2004), as this chip represents a bet that has been
measure is positive, otherwise it is negative.
placed by default. Thus one can interpret a positive signi?cant coef?-
cient on this variable as evidence for these biases in this dataset. A pos-
11 Based on the work of Ethier (1982), Keren and Lewis (1994) report
itive coef?cient on this variable is also consistent with Wagenaar (1988)
that the number of observations necessary to detect a favorable number
who found 70 out of 75 winning bets in his data were not moved. How-
(bias) is generally quite large. For example, on a wheel with 37 numbers
ever, as this is not the main focus of our paper, we do not provide a
it would be necessary to view 30,195 spins in order to detect a bias of
lengthy discussion of this ?nding. Interested readers are encouraged to
1/33 with a 90% level of certainty.
contact the author for further discussion.

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Judgment and Decision Making, Vol. 1, No. 1, July 2006
Biases in casino betting
7
Table 1: Spin outcomes and player bets
Frequency Percent
Percent
Outcome Frequency Percent
Outcome outcome outcome expected ?expected
bet
bet
0/0
22
0.024
0.026
?0.002
354
0.016
0
25
0.028
0.026
0.001
442
0.020
1
23
0.025
0.026
?0.001
362
0.016
2
30
0.033
0.026
0.007
450
0.020
3
28
0.031
0.026
0.005
357
0.016
4
15
0.017
0.026
?0.010
375
0.017
5
28
0.031
0.026
0.005
636
0.028
6
20
0.022
0.026
?0.004
363
0.016
7
15
0.017
0.026
?0.010
682
0.030
8
26
0.029
0.026
0.002
633
0.028
9
23
0.025
0.026
?0.001
503
0.022
10
24
0.027
0.026
0.000
484
0.021
11
26
0.029
0.026
0.002
783
0.035
12
21
0.023
0.026
?0.003
360
0.016
13
21
0.023
0.026
?0.003
525
0.023
14
27
0.030
0.026
0.004
649
0.029
15
27
0.030
0.026
0.004
340
0.015
16
25
0.028
0.026
0.001
643
0.029
17
23
0.025
0.026
?0.001
1079
0.048
18
23
0.025
0.026
?0.001
518
0.023
19
30
0.033
0.026
0.007
595
0.026
20
24
0.027
0.026
0.000
983
0.044
21
26
0.029
0.026
0.002
447
0.020
22
32
0.035
0.026
0.009
576
0.026
23
24
0.027
0.026
0.000
746
0.033
24
18
0.020
0.026
?0.006
461
0.020
25
19
0.021
0.026
?0.005
521
0.023
26
15
0.017
0.026
?0.010
703
0.031
27
22
0.024
0.026
?0.002
490
0.022
28
25
0.028
0.026
0.001
827
0.037
29
23
0.025
0.026
?0.001
878
0.039
30
33
0.037
0.026
0.010
695
0.031
31
22
0.024
0.026
?0.002
664
0.029
32
29
0.032
0.026
0.006
925
0.041
33
17
0.019
0.026
?0.008
613
0.027
34
29
0.032
0.026
0.006
597
0.027
35
22
0.024
0.026
?0.002
627
0.028
36
22
0.024
0.026
?0.002
641
0.028

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Judgment and Decision Making, Vol. 1, No. 1, July 2006
Biases in casino betting
8
Table 2: Hot outcome results by individual
112 possible
39 logistic models 112 possible 93 linear models
Coef?cient
logistic models
w/o errors
linear models
w/o errors
Negative Signi?cant (GF)
17
9
19
19
Negative Nonsigni?cant (GF)
39
11
36
29
Positive Nonsigni?cant (HO)
37
10
34
25
Positive Signi?cant (HO)
19
9
23
20
Thus our ?nal model is
ni?cant category, 45% in the negative nonsigni?cant cate-
gory, 45% in the positive nonsigni?cant category and 5%
P
in the positive signi?cant category. We compare the ac-
it = ?0 + ?1Hit + ?2Fit + ?3Lit + ?
tual observations with this expected distribution using a
For each gambler we run two GLMs (one logistic and
chi-squared test. We robustly reject the null that the p-
one linear). Of the 139 gamblers in our sample, not all
values were observed by chance (p < .0001 for all four
had placed enough bets to allow us to estimate these mod-
columns). A similar test on only the positive (negative)
els either with or without errors. Table 2 categorizes the
observations yields similar results (p < .0001 for all eight
results of the coef?cient on the hotness measure (?1) for
comparisons).
each individual using a variety of techniques and error
Results from this ?eld study are consistent with pre-
thresholds. Signi?cant coef?cients here represent esti-
vious lab studies demonstrating individual heterogeneity
mates that are signi?cant at the 5% level using a two-
in gambler’s fallacy/hot outcome beliefs. While some
tailed test.
gamblers cannot be classi?ed reliably; those that can are
As Table 2 shows, we observe signi?cant heterogene-
roughly equally split between betting in a fashion consis-
ity in the population. Approximately half of the players
tent with the gambler’s fallacy and the hot outcome. In
in our data (depending which model the reader prefers)
the next subsection we continue our analysis of roulette
can be categorized as gambler’s fallacy players; when a
data by examining the hot hand and stock of luck biases.
number has previously appeared, the probability of their
betting on it decreases. The other half of the players in
3.3 Hot hand vs. stock of luck
our data can be categorized as hot outcome players; when
a number has previously appeared, the probability of their
There is an important conceptual difference between a be-
betting on it increases.
lief in hot outcomes (e.g., hot numbers) and the hot hand
One concern with this analysis, raised by an astute
(e.g., a hot person). Our second set of analyses investi-
referee, is that running so many regressions must result
gates whether individual’s behavior is consistent with hot
in some false positives (or false negatives). To test for
hand beliefs. To do this, we analyze whether gamblers bet
whether simple chance is causing our results, we con-
on more or fewer numbers in response to previous wins
ducted two further analyses. First, we looked at the un-
and losses. Thus, if I’ve won in the past, I am hot and
derlying p-values from the regressions in each column.
more likely to be (more) in the future.
If these values had been generated randomly, we would
We ?rst examine the average number of bets an indi-
expect them to be uniformly distributed between 0 and
vidual places after winning on the previous spin and after
1. We compared the actual p-values to the uniform dis-
losing on the previous spin. If the former is greater than
tribution using the Kolgoromov-Smirnov test. We con-
the latter, we say this person bets consistently with the
?dently reject the null hypothesis that the p-values were
hot hand. If the reverse, we say this person bets consis-
generated by chance for each of the four columns in Ta-
tently with stock of luck. Of our 139 gamblers, 62 bet
ble 2 (p < .01 for all four comparisons). Within each
consistently with the hot hand and 32 with the stock of
column, we run a similar test for the positive signi?-
luck bias. Of the remaining 45 gamblers, 31 of them ei-
cant/nonsigni?cant individuals, and the negative signif-
ther only won or only lost at the table in our sample while
icant/nonsigni?cant individuals. Again, we con?dently
14 played for only one spin of the wheel.
reject the null hypothesis that the p-values were generated
As a second, more formal analysis we run a general
by chance for each (p < .01 for all eight comparisons).
linear model for each individual. The dependent variable
A more discrete analysis examines the existing cate-
is the number of bets placed on spin t and the independent
gorizations. If the results were due to chance, we would
variables include an indicator variable describing whether
expect 5% of the observations to fall in the negative sig-
the individual has won or lost on spin t-1. Table 3 reports

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Judgment and Decision Making, Vol. 1, No. 1, July 2006
Biases in casino betting
9
the number of subjects whose parameter value falls into
each category. Ninety-six subjects could be categorized
Hot hand
in this way without errors.
As with the previous analysis of the gambler’s fallacy
versus the hot outcome, we ?nd signi?cant individual het-
erogeneity in the hot hand/stock of luck biases. Here,
more subjects act consistently with the hot hand bias
(which predicts a positive relationship between previous
Gambler’s fallacy
Hot outcome
wins and number of bets placed) than with the stock of
luck bias (which predicts a negative relationship). Simi-
lar reliability tests as those described above yield similar
results (p < .01 for the Kolgoromov-Smirnov tests and
Hot hand vs. stock of luck
p < .001 for the chi-squared tests).
Table 3: Hot hand results by individual
Stock of luck
96 linear models
Coef?cient
without errors
Hot outcome vs. gambler’s fallacy
Negative Signi?cant (SL)
6
Negative Nonsigni?cant (SL)
37
Figure 3: Relationship between biases
Positive Nonsigni?cant (HH)
41
Positive Signi?cant (HH)
12
consistently with the gambler’s fallacy (betting on num-
bers that haven’t appeared previously), are more likely to
act consistently with the hot hand (increasing the number
3.4 Correlation of Biases
of bets they place after a win). Almost half the subjects
are in this ?rst category, consistent with previous research
Our data allow us to independently characterize individ-
demonstrating both biases in the lab. In contrast, play-
uals as gambler’s fallacy/hot outcome players and as hot
ers who act consistently with the hot outcome (betting on
hand/stock of luck players. A further analysis examines
numbers that have appeared previously), are more likely
the distribution of players over those four types. Table 4
to act consistently with the stock-of-luck bias (decreasing
presents this distribution, categorizing players based on
the number of bets they place after a win).
the general linear models at the individual level reported
This relationship can be seen in Figure 3, below. Here
in Table 2 (the ?nal column) and Table 3 including those
we graph, for each of the 89 individuals characterized in
categorized as directional.13 We exclude 11 players who
Table 5, their regression parameters on the two biases.
are categorized on one dimension and not on another.
What accounts for the pattern of individual beliefs
found in Figure 3? While further research will be neces-
sary to ?esh out the variables underlying these patterns,
Table 4: Relationship between the biases
we propose locus of control as an organizing explanation
for this pattern. Originally developed by Rotter (1964),
Hot outcome Gambler’s fallacy
Zimbardo (1985) de?nes locus of control as: “. . . a belief
Hot Hand
10
42
about whether the outcomes of our actions are contingent
Stock-of-Luck
32
5
on what we do (internal locus of control) or on events out-
side our personal control (external control orientation)”
(p. 275).
A chi-squared test strongly rejects the null hypothesis
of no relationship between the biases (p < .0001). In par-
Generally a person with an internal locus of control at-
ticular, there appears to be a correlation; players who act
tributes outcomes to personal decisions and efforts while
a person with an external locus of control attributes out-
13Other possible categorizations yield qualitatively identical results
comes to chance or other external factors. Applying this
(e.g., using those categorized both signi?cantly and nonsigni?cantly re-
concept to roulette, a person with an internal locus of con-
gardless of error, restricting attention to those categorized signi?cantly,
trol is likely to attribute previous wins to the decisions he
either only without error or all and using the logistic models to catego-
rize the Gambler’s Fallacy/Hot Outcome subjects rather than the linear
made and thus to connect such winning with gambling
models).
skill. If a player has just won because of skill, then these

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Judgment and Decision Making, Vol. 1, No. 1, July 2006
Biases in casino betting
10
skills should lead to more winning, which explains why
ases even in this, sophisticated, population, providing an
players with such beliefs increase their bets after winning,
important robustness check on previous laboratory data.
exhibiting hot hand behavior. On the other hand, a per-
Like this previous research, we observe signi?cant indi-
son with an external locus of control attributes winning to
vidual heterogeneity in the population. Our participants
simply luck. Thus a person with external locus of control
are split almost evenly between betting in a way consis-
concludes that winning again after a previous win is less
tent with the gambler’s fallacy and consistent with the hot
likely and will decrease their bets after winning, exhibit-
outcome.
ing stock of luck behavior.
Importantly, however, our data allow us to investigate
Remember that while the hot hand/stock of luck de-
the correlation of these biases at the individual level. We
scribes beliefs of outcomes of the individual (like wins
?nd that gambler’s fallacy players are more likely to also
and losses), the gambler’s fallacy/hot outcome describes
be hot hand gamblers. These relationships suggest there
beliefs about outcomes of the random process (like heads
may be an underlying construct determining biased be-
or tails). So how would the beliefs of a person with an in-
liefs that further research might illuminate. Candidates
ternal or external locus of control differ regarding random
for this construct have been suggested by us and others
processes?
(locus of control, representativeness, cognitive re?ection
Consider ?rst the person who has an external locus
of Frederick [2005]), but further research in the lab will
of control and thus attributes outcomes to luck (stock of
be need to identify these potential mediators.
luck). If one believes luck is in control of a random pro-
These results are consistent with those previously ob-
cess and three heads in a row have appeared, then one
served in the lab (e.g., Ayton & Fischer, 2004; Chau &
should believe that luck will continue to control the out-
Phillips, 1995). That these observations are robust in the
comes and that another head will appear. Put another
?eld with real money on the line and real participants
way, players who believe in luck are more likely to be-
is reassuring. However, the limitations inherent in ?eld
lieve in streaks (hot outcomes) because luck produces
data admit of alternative interpretations of our results.
streaks. Thus the external locus of control causes both
For example, the hot outcome effect may be explained
stock of luck and hot outcome beliefs.
by an availability bias; individuals are more likely to bet
In contrast, a person with an internal locus of control
on numbers that have recently won not because they be-
who believes that winning is a result of skill is likely to
lieve these numbers more likely to win again but instead
reject the idea that the process producing the outcomes
because they’re easily called to mind. The hot hand ef-
is random since this would mitigate the skill involved.
fect may be explained by an income or house money ef-
A more plausible belief is that outcomes on the roulette
fect; individuals bet on more numbers after they have won
wheel are controlled by some process that can be learned
not because they believe that they (personally) are more
or discerned by the use of skill. When the internal per-
likely to win again but because they’re richer, or are play-
son wins, it is con?rmation that she has ascertained the
ing with the house’s money. While these alternative ex-
pattern and this con?dence leads her to bet more on the
planations can explain some results, they don’t provide
next spin of the wheel (hot hand). The most plausible
satisfactory explanations for the heterogeneity of the data
cognitive explanation for her supposed pattern-detecting
at the individual level, nor for the correlation between the
skill is representativeness, which explains why she bets
biases observed within the individual.
consistently with the gambler’s fallacy. Thus the inter-
These limitations suggest further research combining
nal locus of control causes both hot hand and gambler’s
empirical and questionnaire data in a way that we were
fallacy beliefs.
prevented from accomplishing here.
For example, a
Unfortunately we could not collect locus of control or
think-aloud protocol might provide evidence in favor or
other personality measures from our casino patrons, and
against these alternative explanations. Gathering psycho-
thus cannot test our speculation of the underlying causes
logical measures like locus of control as well as demo-
of the relationship between these two biases. Further lab
graphic information might help us to predict what type of
testing will be necessary to address this question, and to
biased beliefs an individual is likely to have. Finally, our
compare this speculation with other candidate explana-
data infers beliefs from observed actions; eliciting beliefs
tions for our results.
directly via a questionnaire, then observing actions would
provide a useful check on our results. These combina-
tions of ?eld and lab data are attractive, but will require
4 Conclusions and discussion
extreme cooperation from a casino, which is not currently
available.
This paper uses observational data to demonstrate the ex-
Other future projects might involve data from other
istence and impact of the hot hand and gambler’s fallacy
non-autocorrelated casino games (e.g., craps, slot ma-
biases. We demonstrate the existence of signi?cant bi-
chines) both to replicate our current ?ndings and to search

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