Carry on winning: The gamblers’ fallacy creates hot hand effects
in online gambling
, Nigel Harvey
Department of Cognitive, Perceptual and Brain Sciences, University College London, UK
Received 12 October 2012
Revised 16 January 2014
Accepted 17 January 2014
People suffering from the hot-hand fallacy unreasonably expect winning streaks to con-
tinue whereas those suffering from the gamblers’ fallacy unreasonably expect losing
streaks to reverse. We took 565,915 sports bets made by 776 online gamblers in 2010
and analyzed all winning and losing streaks up to a maximum length of six. People who
won were more likely to win again (apparently because they chose safer odds than before)
whereas those who lost were more likely to lose again (apparently because they chose risk-
ier odds than before). However, selection of safer odds after winning and riskier ones after
losing indicates that online sports gamblers expected their luck to reverse: they suffered
from the gamblers’ fallacy. By believing in the gamblers’ fallacy, they created their own
Ó 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC
BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
The hot-hand fallacy and gamblers’ fallacy are assumed
to be common among gamblers because it is thought that
they believe that outcomes for future bets are predictable
from those of previous ones.
1.1. Belief in a hot-hand: ‘‘If you have been winning, you are
more likely to win again.’’
The term a ‘‘hot hand’’ was initially used in basketball to
describe a basketball player who had been very successful
in scoring over a short period. It was believed that such a
player had a ‘‘hot hand’’ and that other players should pass
the ball to him to score more. This term is now used more
generally to describe someone who is winning persistently
and can be regarded as ‘‘in luck’’. In gambling scenarios, a
player with a genuine hot hand should keep betting and
There have been extensive discussions about the exis-
tence of the hot hand effect. Some researchers have failed
to ﬁnd any evidence of such an effect (Gilovich, Vallone,
& Tversky, 1985; Koehler & Conley, 2003; Larkey, Smith,
& Kadane, 1989; Wardrop, 1999). Others claim there is evi-
dence of the hot hand effect in games that require consid-
erable physical skill, such as golf, darts, and basketball
(Arkes, 2010, 2011; Gilden & Wilson, 1995; Yaari &
People gambling on sports outcomes may continue to
do so after winning because they believe they have a hot
hand. Such a belief may be a fallacy. It is, however, possible
that their belief is reasonable. For example, on some occa-
sions, they may realize that their betting strategy is pro-
ducing proﬁts and that it would be sensible to continue
with it. Alternatively, a hot hand could arise from some
change in their betting strategy. For example, after win-
ning, they may modify their bets in some way to increase
their chances of winning again.
0010-0277/Ó 2014 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
Corresponding author. Address: Department of Cognitive, Perceptual
and Brain Sciences, University College London, Gower Street, London
WC1E 6BT, UK. Tel.: +44 (0)2076797570.
E-mail address: email@example.com (J. Xu).
Cognition 131 (2014) 173–180
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1.2. The gamblers’ fallacy: ‘‘If you have been losing, you are
more likely to win in future.’’
People gambling on sports outcomes may continue to
do so after losing because they believe in the gamblers’ fal-
lacy. This is the erroneous belief that deviations from ini-
tial expectations are corrected even when outcomes are
produced by independent random processes. Thus, peo-
ple’s initial expectations that, in the long run, tosses of a
fair coin will result in a 50:50 chance of heads and tails
are associated with a belief that deviations from that ratio
will be corrected. Hence, if ﬁve tosses of a fair coin have
produced a sequence of ﬁve heads, the chance of tails on
the next toss will be judged to be larger than 50%. This is
because the coin ‘‘ought to’’ have a 50:50 chance of heads
and tails in the long run and, as a result, more tails are
‘‘needed’’ to correct the deviation from that ratio produced
by the ﬁrst ﬁve tosses.
1.3. Odds and stake size: A conﬂict between belief in a hot
hand and the gambler’s fallacy
Betting strategies are often based on the previous bet-
ting results (Oskarsson, Van Boven, McClelland, & Hastie,
2009). The strategies based on a belief in a hot hand and
gamblers’ fallacy may conﬂict. For example, when trying
to decide what odds to select in the next round, a belief
in the gamblers’ fallacy would result in betting on higher
odds and with more money after losing than after winning.
A believer in the hot hand would do the opposite.
2. Method and data
To date, there is little research on real gambling. Our re-
search (1) demonstrates the existence of a hot hand, (2)
investigates gamblers’ beliefs in a hot hand and the gam-
blers’ fallacy, and (3) explores the causal relationship
between a hot hand and the gamblers’ fallacy.
2.1. Analysis methods
We used a large online gambling database. First, we
counted all the sports betting results to see whether win-
ning was more likely after a streak of winning bets or after
a streak of losing ones. Second, we examined the record of
those gamblers who has long streaks of wins to see
whether they had higher returns; this could be a sign of
real skill. Third, we used the odds and the stake size to
predict the probability of winning.
2.2. Data set
The complete gambling history of 776 gamblers be-
tween 1 January 2010 and 31 December 2010 was ob-
tained from an online gambling company. In total,
565,915 bets were placed by these gamblers during the
year. Characteristics of the samples are shown in Table 1.
Each gambling record included the following informa-
tion: game type (e.g., horse racing, football, and cricket),
game name (e.g. Huddersﬁeld v West Bromwich), time,
stake, type of bet, odds, result, and payoff. Each person
was identiﬁed by a unique account number. All the bets
they placed in the year were arranged in chronological or-
der by the time of settlement, which was precise to the
minute. The time when the stake was placed was not avail-
able but, according to the gambling house, there is no rea-
son to think that stakes are placed long before the time of
settlement. Each account used one currency, which was
chosen when the account was opened; no change of cur-
rency was allowed during the year.
If there is a hot hand, then, after a winning bet, the
probability of winning the next bet should go up. We com-
pared the probability of winning after different run lengths
of previous wins (Fig. 1). If the gamblers’ fallacy is not a fal-
lacy, the probability of winning should go up after losing
several bets. We also compared the probability of winning
in this situation.
3. Results and analysis
3.1. The hot hand
To produce the top panel of Fig. 1, we ﬁrst counted all
the bets in GBP; there were 178,947 bets won and
192,359 bets lost. The probability of winning was 0.48.
Second, we took all the 178,947winning bets and
counted the number of bets that won again; there were
88,036 bets won. The probability of winning was 0.49. In
comparison, following the 192,359 lost bets, the probability
of winning was 0.47. The probability of winning in these two
situations was signiﬁcantly different (Z = 12.10, p < .0001).
Third, we took all the 88,036 bets, which had already
won twice and examined the results of bets that followed
these bets. There were 50,300 bets won. The probability
of winning rose to 0.57. In contrast, the probability of
winning did not rise after gambles that did not show a
winning streak: it was 0.45. The probability of winning
in these two situations was signiﬁcantly different
(Z = 60.74, p < .0001).
Fourth, we examined the 50,300 bets which had already
won three times and checked the result of the bets followed
them. We found that 33,871 bets won. The probability of
winning went up again to 0.67. In contrast, the bets not hav-
ing a run of lucky predecessors showed a probability of win-
ning of 0.45. The probability of winning in these two
situations was signiﬁcantly different (Z = 90.63, p < .0001).
Fifth, we used the same procedure and took all the
33,871 bets which had already won four times. We
checked the result of bets followed these bets. There were
24,390 bets that won. The probability of winning went up
again to 0.72. In contrast, the bets without a run of previ-
ous wins showed a probability of winning of only 0.45.
The probability of winning in these two situations was sig-
niﬁcantly different (Z = 91.96, p < .0001).
Sixth, we used the same method to check the 24,390
bets which had already won ﬁve times in a row. There
were 18,190 bets that won, giving a probability of winning
of 0.75. After other bets, the probability of winning was
0.46. The probability of winning in these two cases was sig-
niﬁcantly different (Z = 86.78, p < .0001).
174 J. Xu, N. Harvey / Cognition 131 (2014) 173–180
Seventh, we examined the 18,190 bets that had won
six times in a row. Following such a lucky streak, the
probability of winning was 0.76. However, for the bets
that had not won on the immediately preceding
Sample characteristics for sports bets placed in each of three currencies for the year 2010.
GBP EUR USD
Number of bets 371,306 162,077 32,532
Number of gamblers 407 318 51
Mean stake £145 (1482) €395 (5555) $50 (321)
Median stake £14 €18 $15
Maximum stake £313,900 €1,492,000 $20,500
Mean number of bets placed by a single account 917 517 641
Median number of bets placed by a single account 171 88 153
Number of horse racing bets 260,550 34,659 8290
Number of soccer bets 69,863 90,415 12,058
Number of greyhound racing bets 28,859 6660 9159
Fig. 1. Probability of winning after obtaining winning streaks of different lengths (o) and after not obtaining winning streaks of those lengths (
J. Xu, N. Harvey / Cognition 131 (2014) 173–180
occasion, the probability of winning was only 0.47. These
two probabilities of winning were signiﬁcantly different
(Z = 77.50, p < .0001).
The hot hand also occurred for bets in other currencies
(Fig. 1). Regressions (Table 2) show that, after each succes-
sive winning bet, the probability of winning increased by
0.05 (t(5) = 8.90, p < .001) for GBP, by 0.06 for EUR
(t(5) = 8.00, p < .001), and by 0.05 for USD (t(5) = 8.90,
p < .001).
3.2. The gamblers’ fallacy
We used the same approach to analyze the gamblers’
fallacy. The ﬁrst step was same as in the analysis of the
hot hand. We counted all the bets in GBP; there were
178,947 bets won and 192,359 bets lost. The probability
of winning was 0.48 (Fig. 2, top panel).
In the second step, we identiﬁed the 192,359 bets that
lost and examined results of the bets immediately after
them. Of these, 90,764 won and 101,595 lost. The probabil-
ity of winning was 0.47. After the 178,947 bets that won,
the probability of winning was 0.49. The difference be-
tween these two probabilities were signiﬁcant (Z = 12.01,
p < 0.001).
In the third step, we took the 101,595 bets that lost and
examined the bets following them. We found that 40,856
bets won and 60,739 bets lost. The probability of winning
after having lost twice was 0.40. In contrast, for the bets
that did not lose on both of the previous rounds, the prob-
ability of winning was 0.51. The difference between these
probabilities was signiﬁcant (Z = 58.63, p < 0.001).
In the fourth step, we repeated the same procedure.
After the 60,739 bets that had lost three times in a row,
there were 19,142 winning bets won and losing 41,595
bets ones, giving a probability of winning of 0.32. For other
bets, this probability was 0.51 (Z = 88.26, p < 0.001).
The ﬁfth, sixth and seventh steps were carried out in an
analogous way. They showed that the probability of win-
ning after four lost bets was 0.27, after ﬁve lost bets was
0.25, and after six lost bets was 0.23.
The pattern was similar for bets in other currencies
(Fig. 2). Regressions (Table 2) showed that each succes-
sive losing bet decreased the probability of winning 0.05
(t(5) = 9.71, p < .001) for GBP, by 0.05 for EUR
(t(5) = 9.10, p < .001) and by 0.02 for USD (t(5) = 7.56,
p < .001). This is bad news for those who believe in the
3.3. Do gamblers with long winning streaks have higher
One potential explanation for the appearance of the hot
hand is that gamblers with long winning streaks consis-
tently do better than others. To examine this possibility,
we compared the mean payoff of these gamblers with
the mean payoff of the remaining gamblers.
Among 407 gamblers using GBP, 144 of them had at
least six successive wins in a row on at least one occasion.
They had a mean loss of £1.0078 (N = 279,162, SD = 0.47)
for every £1 stake they placed. The remaining 263 gam-
blers had a mean loss of £1.0077 (N = 92,144, SD = 0.38)
for every £1 stake they placed. The difference between
these two was not signiﬁcant.
We did same analysis for bets made in EUR. Among 318
gamblers using this currency, 111 of them had at least one
winning streak of six. They had a mean loss of €1.005
(N = 105,136, SD = 0.07) for every €1 of stake. The remain-
ing 207 EUR gamblers had a mean loss of €1.002
(N = 56,941, SD = 0.22). The difference between these two
returns was signiﬁcant (t (162,075) = 4.735, p < 0.0001).
Those who had long winner streaks actually lost more than
The results in USD were similar. Seventeen gamblers
had at least one winning streak of six and 34 did not. For
those who had, the mean loss was $1.022 (N = 23,280,
SD = 0.75); for those who had not, it was $1.029
(N = 9,252, SD = 0.35). There was no signiﬁcant difference
between the two (t (32,530) = 0.861, p = 0.389). The gam-
blers who had long winning streaks were not better at win-
ning money than gamblers who did not have them.
3.4. The effects of winning and losing streaks on level of odds
To determine whether the gamblers believed in the hot
hand or gamblers’ fallacy, we examined how the results of
their gambling affected the odds of their next bet. Among
all GBP gamblers, the mean level of selected odds was
7.72 (N = 371,306, SD = 37.73). After a winning bet, lower
odds were chosen for the next bet. The mean odds dropped
to 6.19 (N = 178,947, SD = 35.02). Following two consecu-
tive winning bets, the mean odds decreased to 3.60
(N = 88,036, SD = 24.69). People who had won on more
consecutive occasions selected less risky odds. This trend
continued (Fig. 3, top panel).
Regression for length of streaks predicting the probability of winning.
BSEBb t Sig. (p) FR
Winning streak 0.475 0.021 0.053 (0.006) 8.902 <0.001 79.25 0.928
Losing streak 0.489 0.018 0.047 (0.004) 9.711 <0.001 94.31 0.940
Winning streak 0.439 0.026 0.059 (0.007) 8.223 <0.001 67.62 0.917
Losing streak 0.508 0.021 0.053 (0.006) 9.100 <0.001 82.8 0.932
Winning streak 0.315 0.025 0.054 (0.007) 7.996 <0.001 63.93 0.913
Losing streak 0.386 0.010 0.022 (0.003) 7.560 <0.001 57.15 0.904
Note: Independent variable is the number of bets taken into consideration.
176 J. Xu, N. Harvey / Cognition 131 (2014) 173–180
After a losing bet, the opposite was found. People who
had lost on more consecutive occasions selected riskier
odds. After six lost bets in a row, the mean odds went up
to 17.07 (N = 22,694, SD = 50.62). In comparison, after win-
ning six times in a row, the ﬁgure for mean odds was 0.85
(N = 18,252, SD = 9.82). From the odds that they selected,
we can infer that gamblers believed in the gamblers’ fal-
lacy but not in the hot hand.
The gambling results were affected by the gamblers’
choice of odds. One point of odds increase reduced the
probability of winning by 0.035 (SD = 0.003, t(36) =
13.403, p < .001).
3.5. The effects of winning and losing streaks on stake size
Among all GBP gamblers, the median stake was £14
(N = 371,306, Interquartile Rang = 4.80–53.29). After
winning once, the median stake went up to £18.47
(N = 178,947, Interquartile Range = 5.04–66.00). After win-
ning twice in a row, the median stake rose to £20.45
(N = 88,036, Interquartile Range = 8.00–80.00) (Fig. 4, top
For the losing side, the opposite was found. People who
had lost on more consecutive occasions decreased stakes.
After losing once, the median stake went down to £10.89
(N = 192,359, Interquartile Range = 4.00–44.16). In
comparison, after losing twice in a row, the median stake
dropped to £10.00 (N = 101,595, Interquartile
Range = 3.33–30.00). These trends continued (Fig. 4, top
Gamblers increased stake size after winning and
decreased stake size after losing. This could be the result
of more money available after winning and less money
available after losing.
Fig. 2. Probability of winning after obtaining losing streaks of different lengths (o) and after not obtaining losing streaks of those lengths (
J. Xu, N. Harvey / Cognition 131 (2014) 173–180
We examined EUR and USD bets. Findings for selected
odds were similar (Fig. 3) but those for stake size were less
robust (Fig. 4), perhaps because of the reduced sample size.
We found evidence for the hot hand but not for the
gamblers’ fallacy. Gamblers were more likely to win after
winning and to lose after losing.
After winning, gamblers selected safer odds. After los-
ing, they selected riskier odds. After winning or losing, they
expected the trend to reverse: they believed the gamblers’
fallacy. However, by believing in the gamblers’ fallacy, peo-
ple created their own luck. The result is ironic: Winners
worried their good luck was not going to continue, so they
selected safer odds. By doing so, they became more likely
to win. The losers expected the luck to turn, so they took
riskier odds. However, this made them even more likely
to lose. The gamblers’ fallacy created the hot hand.
Ayton and Fischer (2004) found that people believed in
the gamblers’ fallacy for natural events over which they
had no control. Our gamblers displayed the gamblers’ fal-
lacy for actions (i.e. bets) that they took themselves. This
may indicate that they did not believe that bets were under
their control. Fong, Law, and Lam (2013) reported Chinese
gamblers believed their luck would continue. Does this
mean they felt they had more control over their bets? By
believing their luck would continue, did they help to bring
it to an end?
Fig. 3. Mean preferred odds after winning (o) and losing (
) streaks of different lengths.
178 J. Xu, N. Harvey / Cognition 131 (2014) 173–180
There are likely to be other domains (e.g., ﬁnancial trad-
ing) where people reduce their preference for risk in the
wake of chance success and thereby give the impression
of a hot hand. Furthermore, they may attribute their suc-
cesses to skill rather than chance (Langer, 1975) and may
not be aware of their change in risk preference. In such cir-
cumstances, they may develop the illusion that they are
becoming better at the task and able to persuade others
that this is so. In the ﬁnancial domain, this would have
clear implications for people’s selection of investment
This research was supported by a scholarship awarded
by the Responsible Gambling Fund to Juemin Xu. We thank
Peter Ayton for invaluable comments on earlier drafts of
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