The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances,
is the mistaken belief that if something happens more frequently than
normal during a given period, it will happen less frequently in the
future (or vice versa). In situations where the outcome being observed
is truly random and consists of independent trials of a random process, this belief is false. The fallacy can arise in many situations, but is most strongly associated with gambling, where it is common among players.
The term "Monte Carlo fallacy" originates from the best known example of the phenomenon, which occurred in the Monte Carlo Casino in 1913.
The term "Monte Carlo fallacy" originates from the best known example of the phenomenon, which occurred in the Monte Carlo Casino in 1913.
Examples
Coin toss
The gambler's fallacy can be illustrated by considering the repeated toss of a fair coin. The outcomes in different tosses are statistically independent and the probability of getting heads on a single toss is 1/2 (one in two). The probability of getting two heads in two tosses is 1/4 (one in four) and the probability of getting three heads in three tosses is 1/8 (one in eight). In general, if Ai is the event where toss i of a fair coin comes up heads, then:
- .
If after tossing four heads in a row, the next coin toss also came up
heads, it would complete a run of five successive heads. Since the
probability of a run of five successive heads is 1/32
(one in thirty-two), a person might believe that the next flip would be
more likely to come up tails rather than heads again. This is incorrect
and is an example of the gambler's fallacy. The event "5 heads in a
row" and the event "first 4 heads, then a tails" are equally likely,
each having probability 1/32. Since the first four tosses turn up heads, the probability that the next toss is a head is:
- .
While a run of five heads has a probability of 1/32 = 0.03125 (a little over 3%), the misunderstanding lies in not realizing that this is the case only before the first coin is tossed.
After the first four tosses, the results are no longer unknown, so
their probabilities are at that point equal to 1 (100%). The reasoning
that it is more likely that a fifth toss is more likely to be tails
because the previous four tosses were heads, with a run of luck in the
past influencing the odds in the future, forms the basis of the fallacy.
Why the probability is 1/2 for a fair coin
If
a fair coin is flipped 21 times, the probability of 21 heads is 1 in
2,097,152. The probability of flipping a head after having already
flipped 20 heads in a row is 1/2. This is an application of Bayes' theorem.
This can also be shown without knowing that 20 heads have occurred, and without applying Bayes' theorem. Assuming a fair coin:
- The probability of 20 heads, then 1 tail is 0.520 × 0.5 = 0.521
- The probability of 20 heads, then 1 head is 0.520 × 0.5 = 0.521
The probability of getting 20 heads then 1 tail, and the probability
of getting 20 heads then another head are both 1 in 2,097,152. When
flipping a fair coin 21 times, the outcome is equally likely to be 21
heads as 20 heads and then 1 tail. These two outcomes are equally as
likely as any of the other combinations that can be obtained from 21
flips of a coin. All of the 21-flip combinations will have probabilities
equal to 0.521, or 1 in 2,097,152. Assuming that a change in
the probability will occur as a result of the outcome of prior flips is
incorrect because every outcome of a 21-flip sequence is as likely as
the other outcomes. In accordance with Bayes' theorem, the likely
outcome of each flip is the probability of the fair coin, which is 1/2.
Other examples
The
fallacy leads to the incorrect notion that previous failures will
create an increased probability of success on subsequent attempts. For a
fair 16-sided die, the probability of each outcome occurring is 1/16 (6.25%). If a win is defined as rolling a 1, the probability of a 1 occurring at least once in 16 rolls is:
The probability of a loss on the first roll is 15/16
(93.75%). According to the fallacy, the player should have a higher
chance of winning after one loss has occurred. The probability of at
least one win is now:
By losing one toss, the player's probability of winning drops by two
percentage points. With 5 losses and 11 rolls remaining, the probability
of winning drops to around 0.5 (50%). The probability of at least one
win does not increase after a series of losses; indeed, the probability
of success actually decreases, because there are fewer trials
left in which to win. The probability of winning will eventually equal
the probability of winning a single toss, which is 1/16 (6.25%) and occurs when only one toss is left.
Reverse position
After
a consistent tendency towards tails, a gambler may also decide that
tails has become a more likely outcome. This is a rational and Bayesian
conclusion, bearing in mind the possibility that the coin may not be
fair; it is not a fallacy. Believing the odds to favor tails, the
gambler sees no reason to change to heads. However it is a fallacy that a
sequence of trials carries a memory of past results which tend to favor
or disfavor future outcomes.
The inverse gambler's fallacy described by Ian Hacking
is a situation where a gambler entering a room and seeing a person
rolling a double six on a pair of dice may erroneously conclude that the
person must have been rolling the dice for quite a while, as they would
be unlikely to get a double six on their first attempt.
Retrospective gambler's fallacy
Researchers
have examined whether a similar bias exists for inferences about
unknown past events based upon known subsequent events, calling this the
"retrospective gambler's fallacy".
An example of a retrospective gambler's fallacy would be to
observe multiple successive "heads" on a coin toss and conclude from
this that the previously unknown flip was "tails". Real world examples of retrospective gambler's fallacy have been argued to exist in events such as the origin of the Universe. In his book Universes,
John Leslie argues that "the presence of vastly many universes very
different in their characters might be our best explanation for why at
least one universe has a life-permitting character". Daniel M. Oppenheimer
and BenoƮt Monin argue that "In other words, the 'best explanation' for
a low-probability event is that it is only one in a multiple of trials,
which is the core intuition of the reverse gambler's fallacy."
Philosophical arguments are ongoing about whether such arguments are or
are not a fallacy, arguing that the occurrence of our universe says
nothing about the existence of other universes or trials of universes.
Three studies involving Stanford University students tested the
existence of a retrospective gamblers' fallacy. All three studies
concluded that people have a gamblers' fallacy retrospectively as well
as to future events.
The authors of all three studies concluded their findings have
significant "methodological implications" but may also have "important
theoretical implications" that need investigation and research, saying
"[a] thorough understanding of such reasoning processes requires that we
not only examine how they influence our predictions of the future, but
also our perceptions of the past."
Childbirth
In 1796, Pierre-Simon Laplace described in A Philosophical Essay on Probabilities
the ways in which men calculated their probability of having sons: "I
have seen men, ardently desirous of having a son, who could learn only
with anxiety of the births of boys in the month when they expected to
become fathers. Imagining that the ratio of these births to those of
girls ought to be the same at the end of each month, they judged that
the boys already born would render more probable the births next of
girls." The expectant fathers feared that if more sons were born in the
surrounding community, then they themselves would be more likely to have
a daughter. This essay by Laplace is regarded as one of the earliest
descriptions of the fallacy.
After having multiple children of the same sex, some parents may
believe that they are due to have a child of the opposite sex. While the
Trivers–Willard hypothesis
predicts that birth sex is dependent on living conditions, stating that
more male children are born in good living conditions, while more
female children are born in poorer living conditions, the probability of
having a child of either sex is still regarded as near 0.5 (50%).
Monte Carlo Casino
Perhaps the most famous example of the gambler's fallacy occurred in a game of roulette at the Monte Carlo Casino
on August 18, 1913, when the ball fell in black 26 times in a row. This
was an extremely uncommon occurrence: the probability of a sequence of
either red or black occurring 26 times in a row is (18/37)26-1
or around 1 in 66.6 million, assuming the mechanism is unbiased.
Gamblers lost millions of francs betting against black, reasoning
incorrectly that the streak was causing an imbalance in the randomness
of the wheel, and that it had to be followed by a long streak of red.
Non-examples
Non-independent events
The gambler's fallacy does not apply in situations where the probability of different events is not independent. In such cases, the probability of future events can change based on the outcome of past events, such as the statistical permutation
of events. An example is when cards are drawn from a deck without
replacement. If an ace is drawn from a deck and not reinserted, the next
draw is less likely to be an ace and more likely to be of another rank.
The probability of drawing another ace, assuming that it was the first
card drawn and that there are no jokers, has decreased from 4/52 (7.69%) to 3/51 (5.88%), while the probability for each other rank has increased from 4/52 (7.69%) to 4/51 (7.84%). This effect allows card counting systems to work in games such as blackjack.
Bias
In most
illustrations of the gambler's fallacy and the reverse gambler's
fallacy, the trial (e.g. flipping a coin) is assumed to be fair. In
practice, this assumption may not hold. For example, if a coin is
flipped 21 times, the probability of 21 heads with a fair coin is 1 in
2,097,152. Since this probability is so small, if it happens, it may
well be that the coin is somehow biased towards landing on heads, or
that it is being controlled by hidden magnets, or similar. In this case, the smart bet is "heads" because Bayesian inference from the empirical evidence
— 21 heads in a row — suggests that the coin is likely to be biased
toward heads. Bayesian inference can be used to show that when the
long-run proportion of different outcomes is unknown but exchangeable
(meaning that the random process from which the outcomes are generated
may be biased but is equally likely to be biased in any direction) and
that previous observations demonstrate the likely direction of the bias,
the outcome which has occurred the most in the observed data is the
most likely to occur again.
For example, if the a priori probability of a biased coin
is say 1%, and assuming that such a biased coin would come down heads
say 60% of the time, then after 21 heads the probability of a biased
coin has increased to about 32%.
The opening scene of the play Rosencrantz and Guildenstern Are Dead by Tom Stoppard discusses these issues as one man continually flips heads and the other considers various possible explanations.
Changing probabilities
If
external factors are allowed to change the probability of the events,
the gambler's fallacy may not hold. For example, a change in the game
rules might favour one player over the other, improving his or her win
percentage. Similarly, an inexperienced player's success may decrease
after opposing teams learn about and play against their weaknesses. This
is another example of bias.
Psychology
Origins
The gambler's fallacy arises out of a belief in a law of small numbers,
leading to the erroneous belief that small samples must be
representative of the larger population. According to the fallacy,
streaks must eventually even out in order to be representative. Amos Tversky and Daniel Kahneman first proposed that the gambler's fallacy is a cognitive bias produced by a psychological heuristic called the representativeness heuristic,
which states that people evaluate the probability of a certain event by
assessing how similar it is to events they have experienced before, and
how similar the events surrounding those two processes are.
According to this view, "after observing a long run of red on the
roulette wheel, for example, most people erroneously believe that black
will result in a more representative sequence than the occurrence of an
additional red",
so people expect that a short run of random outcomes should share
properties of a longer run, specifically in that deviations from average
should balance out. When people are asked to make up a random-looking
sequence of coin tosses, they tend to make sequences where the
proportion of heads to tails stays closer to 0.5 in any short segment
than would be predicted by chance, a phenomenon known as insensitivity to sample size.
Kahneman and Tversky interpret this to mean that people believe short
sequences of random events should be representative of longer ones. The representativeness heuristic is also cited behind the related phenomenon of the clustering illusion,
according to which people see streaks of random events as being
non-random when such streaks are actually much more likely to occur in
small samples than people expect.
The gambler's fallacy can also be attributed to the mistaken
belief that gambling, or even chance itself, is a fair process that can
correct itself in the event of streaks, known as the just-world hypothesis. Other researchers believe that belief in the fallacy may be the result of a mistaken belief in an internal locus of control.
When a person believes that gambling outcomes are the result of their
own skill, they may be more susceptible to the gambler's fallacy because
they reject the idea that chance could overcome skill or talent.
Variations
Some
researchers believe that it is possible to define two types of
gambler's fallacy: type one and type two. Type one is the classic
gambler's fallacy, where individuals believe that a particular outcome
is due after a long streak of another outcome. Type two gambler's
fallacy, as defined by Gideon Keren and Charles Lewis, occurs when a
gambler underestimates how many observations are needed to detect a
favorable outcome, such as watching a roulette wheel for a length of
time and then betting on the numbers that appear most often. For events
with a high degree of randomness, detecting a bias that will lead to a
favorable outcome takes an impractically large amount of time and is
very difficult, if not impossible, to do.
The two types differ in that type one wrongly assumes that gambling
conditions are fair and perfect, while type two assumes that the
conditions are biased, and that this bias can be detected after a
certain amount of time.
Another variety, known as the retrospective gambler's fallacy,
occurs when individuals judge that a seemingly rare event must come from
a longer sequence than a more common event does. The belief that an
imaginary sequence of die rolls is more than three times as long when a
set of three sixes is observed as opposed to when there are only two
sixes. This effect can be observed in isolated instances, or even
sequentially. Another example would involve hearing that a teenager has unprotected sex
and becomes pregnant on a given night, and concluding that she has been
engaging in unprotected sex for longer than if we hear she had
unprotected sex but did not become pregnant, when the probability of
becoming pregnant as a result of each intercourse is independent of the
amount of prior intercourse.
Relationship to hot-hand fallacy
Another psychological perspective states that gambler's fallacy can be seen as the counterpart to basketball's hot-hand fallacy,
in which people tend to predict the same outcome as the previous event -
known as positive recency - resulting in a belief that a high scorer
will continue to score. In the gambler's fallacy, people predict the
opposite outcome of the previous event - negative recency - believing
that since the roulette wheel has landed on black on the previous six
occasions, it is due to land on red the next. Ayton and Fischer have
theorized that people display positive recency for the hot-hand fallacy
because the fallacy deals with human performance, and that people do not
believe that an inanimate object can become "hot."
Human performance is not perceived as random, and people are more
likely to continue streaks when they believe that the process generating
the results is nonrandom.
When a person exhibits the gambler's fallacy, they are more likely to
exhibit the hot-hand fallacy as well, suggesting that one construct is
responsible for the two fallacies.
The difference between the two fallacies is also found in
economic decision-making. A study by Huber, Kirchler, and Stockl in 2010
examined how the hot hand and the gambler's fallacy are exhibited in
the financial market. The researchers gave their participants a choice:
they could either bet on the outcome of a series of coin tosses, use an
expert opinion to sway their decision, or choose a risk-free alternative
instead for a smaller financial reward. Participants turned to the
expert opinion to make their decision 24% of the time based on their
past experience of success, which exemplifies the hot-hand. If the
expert was correct, 78% of the participants chose the expert's opinion
again, as opposed to 57% doing so when the expert was wrong. The
participants also exhibited the gambler's fallacy, with their selection
of either heads or tails decreasing after noticing a streak of either
outcome. This experiment helped bolster Ayton and Fischer's theory that
people put more faith in human performance than they do in seemingly
random processes.
Neurophysiology
While the representativeness heuristic
and other cognitive biases are the most commonly cited cause of the
gambler's fallacy, research suggests that there may also be a
neurological component. Functional magnetic resonance imaging
has shown that after losing a bet or gamble, known as riskloss, the
frontoparietal network of the brain is activated, resulting in more
risk-taking behavior. In contrast, there is decreased activity in the amygdala, caudate, and ventral striatum
after a riskloss. Activation in the amygdala is negatively correlated
with gambler's fallacy, so that the more activity exhibited in the
amygdala, the less likely an individual is to fall prey to the gambler's
fallacy. These results suggest that gambler's fallacy relies more on
the prefrontal cortex, which is responsible for executive, goal-directed
processes, and less on the brain areas that control affective decision-making.
The desire to continue gambling or betting is controlled by the striatum,
which supports a choice-outcome contingency learning method. The
striatum processes the errors in prediction and the behavior changes
accordingly. After a win, the positive behavior is reinforced and after a
loss, the behavior is conditioned to be avoided. In individuals
exhibiting the gambler's fallacy, this choice-outcome contingency method
is impaired, and they continue to make risks after a series of losses.
Possible solutions
The
gambler's fallacy is a deep-seated cognitive bias and can be very hard
to overcome. Educating individuals about the nature of randomness has
not always proven effective in reducing or eliminating any manifestation
of the fallacy. Participants in a study by Beach and Swensson in 1967
were shown a shuffled deck of index cards with shapes on them, and were
instructed to guess which shape would come next in a sequence. The
experimental group of participants was informed about the nature and
existence of the gambler's fallacy, and were explicitly instructed not
to rely on run dependency to make their guesses. The control group was
not given this information. The response styles of the two groups were
similar, indicating that the experimental group still based their
choices on the length of the run sequence. This led to the conclusion
that instructing individuals about randomness is not sufficient in
lessening the gambler's fallacy.
An individual's susceptibility to the gambler's fallacy may
decrease with age. A study by Fischbein and Schnarch in 1997
administered a questionnaire to five groups: students in grades 5, 7, 9,
11, and college students specializing in teaching mathematics. None of
the participants had received any prior education regarding probability.
The question asked was: "Ronni flipped a coin three times and in all
cases heads came up. Ronni intends to flip the coin again. What is the
chance of getting heads the fourth time?" The results indicated that as
the students got older, the less likely they were to answer with
"smaller than the chance of getting tails", which would indicate a
negative recency effect. 35% of the 5th graders, 35% of the 7th graders,
and 20% of the 9th graders exhibited the negative recency effect. Only
10% of the 11th graders answered this way, and none of the college
students did. Fischbein and Schnarch theorized that an individual's
tendency to rely on the representativeness heuristic and other cognitive biases can be overcome with age.
Another possible solution comes from Roney and Trick, Gestalt
psychologists who suggest that the fallacy may be eliminated as a
result of grouping. When a future event such as a coin toss is described
as part of a sequence, no matter how arbitrarily, a person will
automatically consider the event as it relates to the past events,
resulting in the gambler's fallacy. When a person considers every event
as independent, the fallacy can be greatly reduced.
Roney and Trick told participants in their experiment that they
were betting on either two blocks of six coin tosses, or on two blocks
of seven coin tosses. The fourth, fifth, and sixth tosses all had the
same outcome, either three heads or three tails. The seventh toss was
grouped with either the end of one block, or the beginning of the next
block. Participants exhibited the strongest gambler's fallacy when the
seventh trial was part of the first block, directly after the sequence
of three heads or tails. The researchers pointed out that the
participants that did not show the gambler's fallacy showed less
confidence in their bets and bet fewer times than the participants who
picked with the gambler's fallacy. When the seventh trial was grouped
with the second block, and was perceived as not being part of a streak,
the gambler's fallacy did not occur.
Roney and Trick argued that instead of teaching individuals about
the nature of randomness, the fallacy could be avoided by training
people to treat each event as if it is a beginning and not a
continuation of previous events. They suggested that this would prevent
people from gambling when they are losing, in the mistaken hope that
their chances of winning are due to increase based on an interaction
with previous events.
Users
Studies
have found that asylum judges, loan officers, baseball umpires and lotto
players employ the gambler's fallacy consistently in their
decision-making.