Regression toward the mean

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Regression_toward_the_mean

Galton's experimental setup (Fig.8)

In statistics, regression toward the mean (also called reversion to the mean, and reversion to mediocrity) is the phenomenon where if one sample of a random variable is extreme, the next sampling of the same random variable is likely to be closer to its mean. Furthermore, when many random variables are sampled and the most extreme results are intentionally picked out, it refers to the fact that (in many cases) a second sampling of these picked-out variables will result in "less extreme" results, closer to the initial mean of all of the variables.

Mathematically, the strength of this "regression" effect is dependent on whether or not all of the random variables are drawn from the same distribution, or if there are genuine differences in the underlying distributions for each random variable. In the first case, the "regression" effect is statistically likely to occur, but in the second case, it may occur less strongly or not at all.

Regression toward the mean is thus a useful concept to consider when designing any scientific experiment, data analysis, or test, which intentionally selects the most extreme events - it indicates that follow-up checks may be useful in order to avoid jumping to false conclusions about these events; they may be genuine extreme events, a completely meaningless selection due to statistical noise, or a mix of the two cases.

Conceptual examples

Simple example: students taking a test

Consider a class of students taking a 100-item true/false test on a subject. Suppose that all students choose randomly on all questions. Then, each student's score would be a realization of one of a set of independent and identically distributed random variables, with an expected mean of 50. Naturally, some students will score substantially above 50 and some substantially below 50 just by chance. If one selects only the top scoring 10% of the students and gives them a second test on which they again choose randomly on all items, the mean score would again be expected to be close to 50. Thus the mean of these students would "regress" all the way back to the mean of all students who took the original test. No matter what a student scores on the original test, the best prediction of their score on the second test is 50.

If choosing answers to the test questions was not random – i.e. if there were no luck (good or bad) or random guessing involved in the answers supplied by the students – then all students would be expected to score the same on the second test as they scored on the original test, and there would be no regression toward the mean.

Most realistic situations fall between these two extremes: for example, one might consider exam scores as a combination of skill and luck. In this case, the subset of students scoring above average would be composed of those who were skilled and had not especially bad luck, together with those who were unskilled, but were extremely lucky. On a retest of this subset, the unskilled will be unlikely to repeat their lucky break, while the skilled will have a second chance to have bad luck. Hence, those who did well previously are unlikely to do quite as well in the second test even if the original cannot be replicated.

The following is an example of this second kind of regression toward the mean. A class of students takes two editions of the same test on two successive days. It has frequently been observed that the worst performers on the first day will tend to improve their scores on the second day, and the best performers on the first day will tend to do worse on the second day. The phenomenon occurs because student scores are determined in part by underlying ability and in part by chance. For the first test, some will be lucky, and score more than their ability, and some will be unlucky and score less than their ability. Some of the lucky students on the first test will be lucky again on the second test, but more of them will have (for them) average or below average scores. Therefore, a student who was lucky and over-performed their ability on the first test is more likely to have a worse score on the second test than a better score. Similarly, students who unluckily score less than their ability on the first test will tend to see their scores increase on the second test. The larger the influence of luck in producing an extreme event, the less likely the luck will repeat itself in multiple events.

Other examples

If your favourite sports team won the championship last year, what does that mean for their chances for winning next season? To the extent this result is due to skill (the team is in good condition, with a top coach, etc.), their win signals that it is more likely they will win again next year. But the greater the extent this is due to luck (other teams embroiled in a drug scandal, favourable draw, draft picks turned out to be productive, etc.), the less likely it is they will win again next year.

If a business organisation has a highly profitable quarter, despite the underlying reasons for its performance being unchanged, it is likely to do less well the next quarter.

Baseball players who hit well in their rookie season are likely to do worse their second; the "sophomore slump". Similarly, regression toward the mean is an explanation for the Sports Illustrated cover jinx — periods of exceptional performance which results in a cover feature are likely to be followed by periods of more mediocre performance, giving the impression that appearing on the cover causes an athlete's decline.

History

Discovery

Francis Galton's 1886 illustration of the correlation between the heights of adults and their parents. The observation that adult children's heights tended to deviate less from the mean height than their parents suggested the concept of "regression toward the mean", giving regression analysis its name.

The concept of regression comes from genetics and was popularized by Sir Francis Galton during the late 19th century with the publication of Regression towards mediocrity in hereditary stature. Galton observed that extreme characteristics (e.g., height) in parents are not passed on completely to their offspring. Rather, the characteristics in the offspring regress toward a mediocre point (a point which has since been identified as the mean). By measuring the heights of hundreds of people, he was able to quantify regression to the mean, and estimate the size of the effect. Galton wrote that, "the average regression of the offspring is a constant fraction of their respective mid-parental deviations". This means that the difference between a child and its parents for some characteristic is proportional to its parents' deviation from typical people in the population. If its parents are each two inches taller than the averages for men and women, then, on average, the offspring will be shorter than its parents by some factor (which, today, we would call one minus the regression coefficient) times two inches. For height, Galton estimated this coefficient to be about 2/3: the height of an individual will measure around a midpoint that is two thirds of the parents' deviation from the population average.

Galton also published these results using the simpler example of pellets falling through a Galton board to form a normal distribution centred directly under their entrance point. These pellets might then be released down into a second gallery corresponding to a second measurement. Galton then asked the reverse question: "From where did these pellets come?"

The answer was not 'on average directly above'. Rather it was 'on average, more towards the middle', for the simple reason that there were more pellets above it towards the middle that could wander left than there were in the left extreme that could wander to the right, inwards.

Evolving usage of the term

Galton coined the term "regression" to describe an observable fact in the inheritance of multi-factorial quantitative genetic traits: namely that traits of the offspring of parents who lie at the tails of the distribution often tend to lie closer to the centre, the mean, of the distribution. He quantified this trend, and in doing so invented linear regression analysis, thus laying the groundwork for much of modern statistical modelling. Since then, the term "regression" has been used in other contexts, and it may be used by modern statisticians to describe phenomena such as sampling bias which have little to do with Galton's original observations in the field of genetics.

Galton's explanation for the regression phenomenon he observed in biology was stated as follows: "A child inherits partly from his parents, partly from his ancestors. Speaking generally, the further his genealogy goes back, the more numerous and varied will his ancestry become, until they cease to differ from any equally numerous sample taken at haphazard from the race at large." Galton's statement requires some clarification in light of knowledge of genetics: Children receive genetic material from their parents, but hereditary information (e.g. values of inherited traits) from earlier ancestors can be passed through their parents (and may not have been expressed in their parents). The mean for the trait may be nonrandom and determined by selection pressure, but the distribution of values around the mean reflects a normal statistical distribution.

The population-genetic phenomenon studied by Galton is a special case of "regression to the mean"; the term is often used to describe many statistical phenomena in which data exhibit a normal distribution around a mean.

Importance

Regression toward the mean is a significant consideration in the design of experiments.

Take a hypothetical example of 1,000 individuals of a similar age who were examined and scored on the risk of experiencing a heart attack. Statistics could be used to measure the success of an intervention on the 50 who were rated at the greatest risk, as measured by a test with a degree of uncertainty. The intervention could be a change in diet, exercise, or a drug treatment. Even if the interventions are worthless, the test group would be expected to show an improvement on their next physical exam, because of regression toward the mean. The best way to combat this effect is to divide the group randomly into a treatment group that receives the treatment, and a group that does not. The treatment would then be judged effective only if the treatment group improves more than the untreated group.

Alternatively, a group of disadvantaged children could be tested to identify the ones with most college potential. The top 1% could be identified and supplied with special enrichment courses, tutoring, counseling and computers. Even if the program is effective, their average scores may well be less when the test is repeated a year later. However, in these circumstances it may be considered unethical to have a control group of disadvantaged children whose special needs are ignored. A mathematical calculation for shrinkage can adjust for this effect, although it will not be as reliable as the control group method (see also Stein's example).

The effect can also be exploited for general inference and estimation. The hottest place in the country today is more likely to be cooler tomorrow than hotter, as compared to today. The best performing mutual fund over the last three years is more likely to see relative performance decline than improve over the next three years. The most successful Hollywood actor of this year is likely to have less gross than more gross for his or her next movie. The baseball player with the highest batting average by the All-Star break is more likely to have a lower average than a higher average over the second half of the season.

Misunderstandings

The concept of regression toward the mean can be misused very easily.

In the student test example above, it was assumed implicitly that what was being measured did not change between the two measurements. Suppose, however, that the course was pass/fail and students were required to score above 70 on both tests to pass. Then the students who scored under 70 the first time would have no incentive to do well, and might score worse on average the second time. The students just over 70, on the other hand, would have a strong incentive to study and concentrate while taking the test. In that case one might see movement away from 70, scores below it getting lower and scores above it getting higher. It is possible for changes between the measurement times to augment, offset or reverse the statistical tendency to regress toward the mean.

Statistical regression toward the mean is not a causal phenomenon. A student with the worst score on the test on the first day will not necessarily increase his score substantially on the second day due to the effect. On average, the worst scorers improve, but that is only true because the worst scorers are more likely to have been unlucky than lucky. To the extent that a score is determined randomly, or that a score has random variation or error, as opposed to being determined by the student's academic ability or being a "true value", the phenomenon will have an effect. A classic mistake in this regard was in education. The students that received praise for good work were noticed to do more poorly on the next measure, and the students who were punished for poor work were noticed to do better on the next measure. The educators decided to stop praising and keep punishing on this basis. Such a decision was a mistake, because regression toward the mean is not based on cause and effect, but rather on random error in a natural distribution around a mean.

Although extreme individual measurements regress toward the mean, the second sample of measurements will be no closer to the mean than the first. Consider the students again. Suppose the tendency of extreme individuals is to regress 10% of the way toward the mean of 80, so a student who scored 100 the first day is expected to score 98 the second day, and a student who scored 70 the first day is expected to score 71 the second day. Those expectations are closer to the mean than the first day scores. But the second day scores will vary around their expectations; some will be higher and some will be lower. For extreme individuals, we expect the second score to be closer to the mean than the first score, but for all individuals, we expect the distribution of distances from the mean to be the same on both sets of measurements.

Related to the point above, regression toward the mean works equally well in both directions. We expect the student with the highest test score on the second day to have done worse on the first day. And if we compare the best student on the first day to the best student on the second day, regardless of whether it is the same individual or not, there is no tendency to regress toward the mean going in either direction. We expect the best scores on both days to be equally far from the mean.

Regression fallacies

Main article: Regression fallacy

Many phenomena tend to be attributed to the wrong causes when regression to the mean is not taken into account.

An extreme example is Horace Secrist's 1933 book The Triumph of Mediocrity in Business, in which the statistics professor collected mountains of data to prove that the profit rates of competitive businesses tend toward the average over time. In fact, there is no such effect; the variability of profit rates is almost constant over time. Secrist had only described the common regression toward the mean. One exasperated reviewer, Harold Hotelling, likened the book to "proving the multiplication table by arranging elephants in rows and columns, and then doing the same for numerous other kinds of animals".

The calculation and interpretation of "improvement scores" on standardized educational tests in Massachusetts probably provides another example of the regression fallacy. In 1999, schools were given improvement goals. For each school, the Department of Education tabulated the difference in the average score achieved by students in 1999 and in 2000. It was quickly noted that most of the worst-performing schools had met their goals, which the Department of Education took as confirmation of the soundness of their policies. However, it was also noted that many of the supposedly best schools in the Commonwealth, such as Brookline High School (with 18 National Merit Scholarship finalists) were declared to have failed. As in many cases involving statistics and public policy, the issue is debated, but "improvement scores" were not announced in subsequent years and the findings appear to be a case of regression to the mean.

The psychologist Daniel Kahneman, winner of the 2002 Nobel Memorial Prize in Economic Sciences, pointed out that regression to the mean might explain why rebukes can seem to improve performance, while praise seems to backfire.

I had the most satisfying Eureka experience of my career while attempting to teach flight instructors that praise is more effective than punishment for promoting skill-learning. When I had finished my enthusiastic speech, one of the most seasoned instructors in the audience raised his hand and made his own short speech, which began by conceding that positive reinforcement might be good for the birds, but went on to deny that it was optimal for flight cadets. He said, "On many occasions I have praised flight cadets for clean execution of some aerobatic maneuver, and in general when they try it again, they do worse. On the other hand, I have often screamed at cadets for bad execution, and in general they do better the next time. So please don't tell us that reinforcement works and punishment does not, because the opposite is the case." This was a joyous moment, in which I understood an important truth about the world: because we tend to reward others when they do well and punish them when they do badly, and because there is regression to the mean, it is part of the human condition that we are statistically punished for rewarding others and rewarded for punishing them. I immediately arranged a demonstration in which each participant tossed two coins at a target behind his back, without any feedback. We measured the distances from the target and could see that those who had done best the first time had mostly deteriorated on their second try, and vice versa. But I knew that this demonstration would not undo the effects of lifelong exposure to a perverse contingency.

The regression fallacy is also explained in Rolf Dobelli's The Art of Thinking Clearly.

UK law enforcement policies have encouraged the visible siting of static or mobile speed cameras at accident blackspots. This policy was justified by a perception that there is a corresponding reduction in serious road traffic accidents after a camera is set up. However, statisticians have pointed out that, although there is a net benefit in lives saved, failure to take into account the effects of regression to the mean results in the beneficial effects being overstated.

Statistical analysts have long recognized the effect of regression to the mean in sports; they even have a special name for it: the "sophomore slump". For example, Carmelo Anthony of the NBA's Denver Nuggets had an outstanding rookie season in 2004. It was so outstanding that he could not be expected to repeat it: in 2005, Anthony's numbers had dropped from his rookie season. The reasons for the "sophomore slump" abound, as sports rely on adjustment and counter-adjustment, but luck-based excellence as a rookie is as good a reason as any. Regression to the mean in sports performance may also explain the apparent "Sports Illustrated cover jinx" and the "Madden Curse". John Hollinger has an alternative name for the phenomenon of regression to the mean: the "fluke rule"^{[citation needed]}, while Bill James calls it the "Plexiglas Principle".

Because popular lore has focused on regression toward the mean as an account of declining performance of athletes from one season to the next, it has usually overlooked the fact that such regression can also account for improved performance. For example, if one looks at the batting average of Major League Baseball players in one season, those whose batting average was above the league mean tend to regress downward toward the mean the following year, while those whose batting average was below the mean tend to progress upward toward the mean the following year.

Other statistical phenomena

Regression toward the mean simply says that, following an extreme random event, the next random event is likely to be less extreme. In no sense does the future event "compensate for" or "even out" the previous event, though this is assumed in the gambler's fallacy (and the variant law of averages). Similarly, the law of large numbers states that in the long term, the average will tend toward the expected value, but makes no statement about individual trials. For example, following a run of 10 heads on a flip of a fair coin (a rare, extreme event), regression to the mean states that the next run of heads will likely be less than 10, while the law of large numbers states that in the long term, this event will likely average out, and the average fraction of heads will tend to 1/2. By contrast, the gambler's fallacy incorrectly assumes that the coin is now "due" for a run of tails to balance out.

The opposite effect is regression to the tail, resulting from a distribution with non-vanishing probability density toward infinity.

Definition for simple linear regression of data points

This is the definition of regression toward the mean that closely follows Sir Francis Galton's original usage.

Suppose there are n data points {y_i, x_i}, where i = 1, 2, ..., n. We want to find the equation of the regression line, i.e. the straight line

${\displaystyle y=\alpha +\beta x,}$

which would provide a best fit for the data points. (Note that a straight line may not be the appropriate regression curve for the given data points.) Here the best will be understood as in the least-squares approach: such a line that minimizes the sum of squared residuals of the linear regression model. In other words, numbers α and β solve the following minimization problem:

Find ${\displaystyle \underset{\alpha ,\beta }{min}Q(\alpha ,\beta )}$, where ${\displaystyle Q(\alpha ,\beta )=\sum _{i=1}^n{\hat{\epsilon }}_i^2=\sum _{i=1}^n(y_i-\alpha -\beta x_i{)}^2}$

Using calculus it can be shown that the values of α and β that minimize the objective function Q are

${\displaystyle \begin{array}{rl}& \hat{\beta }=\frac{\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{\sum _{i=1}^n(x_i-\overline{x}{)}^2}=\frac{\overline{xy}-\overline{x}\overline{y}}{\overline{x^2}-{\overline{x}}^2}=\frac{\mathrm{Cov}[x,y]}{\mathrm{Var}[x]}=r_{xy}\frac{s_y}{s_x},\\ & \hat{\alpha }=\overline{y}-\hat{\beta }\overline{x},\end{array}}$

where r_xy is the sample correlation coefficient between x and y, s_x is the standard deviation of x, and s_y is correspondingly the standard deviation of y. Horizontal bar over a variable means the sample average of that variable. For example: ${\displaystyle \overline{xy}=\frac{1}{n}\sum _{i=1}^n{x}_i{y}_i.}$

Substituting the above expressions for ${\displaystyle \hat{\alpha }}$ and ${\displaystyle \hat{\beta }}$ into ${\displaystyle y=\alpha +\beta x,}$ yields fitted values

${\displaystyle \hat{y}=\hat{\alpha }+\hat{\beta }x,}$

which yields

${\displaystyle \frac{\hat{y}-\overline{y}}{s_y}=r_{xy}\frac{x-\overline{x}}{s_x}}$

This shows the role r_xy plays in the regression line of standardized data points.

If −1 < r_xy < 1, then we say that the data points exhibit regression toward the mean. In other words, if linear regression is the appropriate model for a set of data points whose sample correlation coefficient is not perfect, then there is regression toward the mean. The predicted (or fitted) standardized value of y is closer to its mean than the standardized value of x is to its mean.

Definitions for bivariate distribution with identical marginal distributions

Restrictive definition

Let X₁, X₂ be random variables with identical marginal distributions with mean μ. In this formalization, the bivariate distribution of X₁ and X₂ is said to exhibit regression toward the mean if, for every number c > μ, we have

μ ≤ E[X₂ | X₁ = c] < c,

with the reverse inequalities holding for c < μ.

The following is an informal description of the above definition. Consider a population of widgets. Each widget has two numbers, X₁ and X₂ (say, its left span (X₁ ) and right span (X₂)). Suppose that the probability distributions of X₁ and X₂ in the population are identical, and that the means of X₁ and X₂ are both μ. We now take a random widget from the population, and denote its X₁ value by c. (Note that c may be greater than, equal to, or smaller than μ.) We have no access to the value of this widget's X₂ yet. Let d denote the expected value of X₂ of this particular widget. (i.e. Let d denote the average value of X₂ of all widgets in the population with X₁=c.) If the following condition is true:

Whatever the value c is, d lies between μ and c (i.e. d is closer to μ than c is),

then we say that X₁ and X₂ show regression toward the mean.

This definition accords closely with the current common usage, evolved from Galton's original usage, of the term "regression toward the mean". It is "restrictive" in the sense that not every bivariate distribution with identical marginal distributions exhibits regression toward the mean (under this definition).

Theorem

If a pair (X, Y) of random variables follows a bivariate normal distribution, then the conditional mean E(Y|X) is a linear function of X. The correlation coefficient r between X and Y, along with the marginal means and variances of X and Y, determines this linear relationship:

${\displaystyle \frac{E(Y\mid X)-E[Y]}{{\sigma }_y}=r\frac{X-E[X]}{{\sigma }_x},}$

where E[X] and E[Y] are the expected values of X and Y, respectively, and σ_x and σ_y are the standard deviations of X and Y, respectively.

Hence the conditional expected value of Y, given that X is t standard deviations above its mean (and that includes the case where it's below its mean, when t < 0), is rt standard deviations above the mean of Y. Since |r| ≤ 1, Y is no farther from the mean than X is, as measured in the number of standard deviations.

Hence, if 0 ≤ r < 1, then (X, Y) shows regression toward the mean (by this definition).

General definition

The following definition of reversion toward the mean has been proposed by Samuels as an alternative to the more restrictive definition of regression toward the mean above.

Let X₁, X₂ be random variables with identical marginal distributions with mean μ. In this formalization, the bivariate distribution of X₁ and X₂ is said to exhibit reversion toward the mean if, for every number c, we have

μ ≤ E[X₂ | X₁ > c] < E[X₁ | X₁ > c], and

μ ≥ E[X₂ | X₁ < c] > E[X₁ | X₁ < c]

This definition is "general" in the sense that every bivariate distribution with identical marginal distributions exhibits reversion toward the mean, provided some weak criteria are satisfied (non-degeneracy and weak positive dependence as described in Samuels's paper).

Alternative definition in financial usage

Jeremy Siegel uses the term "return to the mean" to describe a financial time series in which "returns can be very unstable in the short run but very stable in the long run." More quantitatively, it is one in which the standard deviation of average annual returns declines faster than the inverse of the holding period, implying that the process is not a random walk, but that periods of lower returns are systematically followed by compensating periods of higher returns, as is the case in many seasonal businesses, for example.

Gambler's fallacy

From Wikipedia, the free encyclopedia

The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the incorrect belief that, if an event (whose occurrences are independent and identically distributed) has occurred more frequently than expected, it is less likely to happen again in the future (or vice versa). The fallacy is commonly associated with gambling, where it may be believed, for example, that the next dice roll is more than usually likely to be six because there have recently been fewer than the expected number of sixes.

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

Over time, the proportion of red/blue coin tosses approaches 50-50, but the difference decreases to zero non-systematically.

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 A_i is the event where toss i of a fair coin comes up heads, then:

${\displaystyle Pr\left(\bigcap _{i=1}^n{A}_i\right)=\prod _{i=1}^{n}Pr(A_i)=\frac{1}{2^n}}$.

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:

${\displaystyle Pr\left(A_5|A_1\cap A_2\cap A_3\cap A_4\right)=Pr\left(A_5\right)=\frac{1}{2}}$.

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 in this example, the results are no longer unknown, so their probabilities are at that point equal to 1 (100%). The probability of a run of coin tosses of any length continuing for one more toss is always 0.5. The reasoning 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. 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:

${\displaystyle 1-{\left[\frac{15}{16}\right]}^{16}=64.39\mathrm{\%}}$

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:

${\displaystyle 1-{\left[\frac{15}{16}\right]}^{15}=62.02\mathrm{\%}}$

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 be equal to 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 Benoît 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. Likewise, after having multiple children of the same sex, some parents may erroneously believe that they are due to have a child of the opposite sex.

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 when 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 card drawn 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

Types of users

Within a real-world setting, numerous studies have uncovered that for various decision makers placed in high stakes scenarios, it is likely they will reflect some degree of strong negative autocorrelation in their judgement.

Asylum judges

In a study aimed at discovering if the negative autocorrelation that exists with the gambler's fallacy existed in the decision made by U.S. asylum judges, results showed that after two successive asylum grants, a judge would be 5.5% less likely to approve a third grant.

Baseball umpires

In the game of baseball, decisions are made every minute. One particular decision made by umpires which is often subject to scrutiny is the 'strike zone' decision. Whenever a batter does not swing, the umpire must decide if the ball was within a fair region for the batter, known as the strike zone. If outside of this zone, the ball does not count towards outing the batter. In a study of over 12,000 games, results showed that umpires are 1.3% less likely to call a strike if the previous two balls were also strikes.

Loan officers

In the decision making of loan officers, it can be argued that monetary incentives are a key factor in biased decision making, rendering it harder to examine the gambler's fallacy effect. However, research shows that loan officers who are not incentivised by monetary gain are 8% less likely to approve a loan if they approved one for the previous client.

Lottery players

The effect of gambler's fallacy on lottery selections, based on studies by Dek Terrell. After winning numbers are drawn, lottery players respond by reducing the number of times they select those numbers in following draws. This effect slowly corrects over time, as players become less affected by the fallacy.

Lottery play and jackpots entice gamblers around the globe, with the biggest decision for hopeful winners being what numbers to pick. While most people will have their own strategy, evidence shows that after a number is selected as a winner in the current draw, the same number will experience a significant drop in selections in the following lottery. A popular study by Charles Clotfelter and Philip Cook investigated this effect in 1991, where they concluded bettors would cease to select numbers immediately after they were selected, ultimately recovering selection popularity within three months. Soon after, a 1994 study was constructed by Dek Terrell to test the findings of Clotfelter and Cook. The key change in Terrell's study was the examination of a pari-mutuel lottery in which, a number selected with lower total wagers placed on it will result in a higher pay-out. While this examination did conclude that players in both types of lotteries exhibited behaviour in-line with the gambler's fallacy theory, those who took part in pari-mutuel betting seemed to be less influenced.

Table 1. Percentage change in numbers selected by lottery players based on Clotfelter, Cook (1991)

		Amount bet by lottery players
Numbers drawn 14 April 1988		Draw day	Days after draw
April	Winner Numbers	0	1	3	7	56
11	244	41	34	24	27	30
12	504	29	20	12	18	15
13	718	28	20	17	19	25
14	323	134	95	79	81	76
15	640	10	20	18	16	20
16	957	30	22	20	24	32
Average percentage of players selecting previously winning numbers compared to day of draw			78%	63%	68%	73%

The effect the of gambler's fallacy can be observed as numbers are chosen far less frequently soon after they are selected as winners, recovering slowly over a two-month period. For example, on the 11th of April 1988, 41 players selected 244 as the winning combination. Three days later only 24 individuals selected 244, a 41.5% decrease. This is the gambler's fallacy in motion, as lottery players believe that the occurrence of a winning combination in previous days will decrease its likelihood of occurring today.

Video game players

Several video games feature the use of loot boxes, a collection of in-game items awarded on opening with random contents set by rarity metrics, as a monetization scheme. Since around 2018, loot boxes have come under scrutiny from governments and advocates on the basis they are akin to gambling, particularly for games aimed at youth. Some games use a special "pity-timer" mechanism, that if the player has opened several loot boxes in a row without obtaining a high-rarity item, subsequent loot boxes will improve the odds of a higher-rate item drop. This is considered to feed into the gambler's fallacy since it reinforces the idea that a player will eventually obtain a high-rarity item (a win) after only receiving common items from a string of previous loot boxes.