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The Gambler’s Fallacy

The origin of roulette is a bit murky but it’s widely accepted the mathematician Blaise Pascal had a hand in the invention in the 17th century.

The origin of roulette is a bit murky but it’s widely accepted the mathematician Blaise Pascal had a hand in the invention in the 17th century.

Feeling Lucky?

Each time a coin is tossed, there is a fifty-fifty chance it will come down heads. It doesn’t matter how many times the coin came down heads before; the odds always remain fifty-fifty. The coin has no memory of previous outcomes, although the coin-flipper does. Believing that past events influence the probability of future ones causes a lot of trouble for gamblers; it also infects many other aspects of life.

The Roulette Wheel

The only way to win consistently in a casino is to own one (unless you are Donald Trump, but that’s another story). So it was that on the night of August 18, 1913, Le Grande Casino of Monte Carlo made an absolute killing.

Crowds gathered around the roulette table after word spread that the ball had dropped into a black slot ten times in a row. Patrons started pushing bets onto red on the table, but still, the ball fell on black.

Monte Carlo Casino in about 1900.

Monte Carlo Casino in about 1900.

As play continued, the bets got larger, until millions were being wagered on each spin of the wheel. Black again! Gamblers were convinced that red must come up on the next go-round. But, that belief defies logic. The odds of the result being black or red are exactly the same with each turn.

Eventually, on the 27th spin, the streak of blacks ended, but by then, fortunes in the neighbourhood of ten million francs had been lost and handed over to the casino.

The gambler’s fallacy is “The mistaken belief that if a certain independent event occurs more frequently than normal during a certain time period, then it’s less likely to occur in the future.”


The Law of Small Numbers

On a roulette wheel, there are 37 pockets; 18 are black, 18 are red, and one is green for the number zero (American-style wheels have two zero pockets). If the wheel is spun a billion times, a fairly accurate level of probabilities will be produced. Not counting the zero slots, the outcome will be very close to 50-50 for black or red.

Back it up to 100 spins, and the odds are probably going to be something like 48-52 either way. With just ten spins, as we’ve shown with the Monte Carlo incident, the probabilities can be wildly inaccurate.

This is where we meet a phenomenon that goes by several names: the law of small numbers, jumping to a conclusion, faulty generalization, or the fallacy of the lonely fact.

Professor Richard Nordquist on explains, “By definition, an argument based on a hasty generalization always proceeds from the particular to the general. It takes a small sample and tries to extrapolate an idea about that sample and apply it to a larger population, and it doesn’t work.”

Those gamblers in Monte Carlo were doing precisely this; they were taking a small sample and assuming past events would influence future ones. They can’t and don’t.

The Reverse Gambler’s Fallacy

Stepping aside from casino games, the illogical application of the gambler’s fallacy pops up in other places. Academics at the National Bureau of Economic Research (NBER) have found the phenomenon in the United States in such diverse fields as refugee asylum cases, major league baseball, and loan applications.

In the way that university professors like to write, they refer to decision-makers exhibiting “negatively auto-correlated decision-making.” Simply stated, people making decisions unconsciously allow their earlier verdicts to influence later ones; this is the reverse of the gambler’s fallacy.

Judges in U.S. asylum-seeking cases are more likely to grant an application if it follows a case in which they denied asylum. The NBER report says, “We estimate judges are up to 3.3 percentage points more likely to reject the current case if they approved the previous case. This translates into two percent of decisions being reversed purely due to the sequencing of past decisions, all else equal.”

Those don’t sound like big numbers, but the result can be catastrophic for those deported because a judge reflexively allowed a previous decision to impact a later case.

The researchers found the same phenomenon at play with bank loan officers, estimating that “five percent of loan decisions would have gone the other way if not for this type of bias.”

And, every baseball hitter knows for certain that umpires routinely make bad calls. The NBER team found there’s some truth to that, writing that major league baseball umpires “call the same pitches in the exact same location differently depending solely on the sequence of previous calls.”

Every baseball hitter knows for certain that umpires routinely make bad calls.

Every baseball hitter knows for certain that umpires routinely make bad calls.

Hot Hand Bias

Gamblers have a tendency to believe in lucky streaks; because I won my last bet, I’m more likely to win my next one. There’s no evidence to support this notion, and researchers found this idea exists in primates other than humans.

Tommy Blanchard has a PhD in brain and cognitive science. He and colleagues at the University of Rochester, New York, studied the behaviour of monkeys. The primates were given two choices, one of which delivered a reward. The BBC reports that “When the correct option was random―the same 50:50 chance as a coin flip―the monkeys still had a tendency to select the previously winning option, as if luck should continue, clumping together in streaks.”

Monkeys are not tutored in probability theory.

Monkeys are not tutored in probability theory.

Of course, monkeys are not tutored in probability theory; they can’t harbour irrational beliefs in the likelihood of an event happening, so something else must be going on. Dr. Blanchard suggests the behaviour springs from an evolutionary advantage that developed as our ancestors foraged for food.

“If you find an apple lying around somewhere,” he told Wired, “chances are you’re going to find other apples nearby.” From this comes the knowledge that food tends to come in clusters, just as gamblers believe that luck comes in clusters.

Research shows that even though people are aware of the gambler’s fallacy, many are still prey to it. One way to avoid falling into the trap is to apply disciplined, critical thinking to all decisions. Another approach is to not gamble.

“People assume that streaks of outcomes will end up ‘evening out’ in order to be considered representative of what an ideal and fair random streak should look like, because they view chance as a fair and self-correcting process.”

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Bonus Factoids

  • The origin of roulette is a bit murky, but it’s widely accepted the mathematician Blaise Pascal had a hand in the invention in the 17th century. Two similar games were called even-odd and roly-poly.
  • Only a player that bets on zero can win if the ball drops into the zero pocket. Anybody else who bet red or black, even or odd, or any other numbers loses. This gives the house a 2.6% edge. American roulette wheels have a double zero slot as well as a single zero one; this gives the house a 5.26% edge.
  • In the casino world, a “Whale” is a high-stakes gambler who bets millions of dollars in a single session. Casinos compete with lavish gifts to attract whales to their premises.
  • In 1992, Archie Karas was broke when he got a $10,000 loan from a friend. In Las Vegas, he used the loan to start a gambling run that, by the start of 1995, had netted him $40 million. By late 1995, he had lost everything playing craps at Binion’s Gambling Hall.


  • “Hasty Generalization (Fallacy).” Richard Nordquist,, September 7, 2019.
  • “The Gambler’s Fallacy – Explained.” Nick Valentine, The Calculator Site, June 23, 2019.
  • “Hot-Hand Bias in Rhesus Monkeys.” Tommy C. Blanchard et al., National Library of Medicine, July 2014.
  • “Monkeys, Like People, Believe in the Hot-Hand Phenomenon.” Mary Bates, Wired, July 10, 2014.
  • “Decision-Making under the Gambler’s Fallacy: Evidence from Asylum Judges, Loan Officers, and Baseball Umpires.” Daniel Chen et al., National Bureau of Economic Research, 2016.
  • “The Gambler’s Fallacy: On the Danger of Misunderstanding Simple Probabilities.”, undated.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2020 Rupert Taylor


Miebakagh Fiberesima from Port Harcourt, Rivers State, NIGERIA. on May 30, 2020:

Hi, I agreed. Thanks.

Charlie Halliday from Scotland on May 29, 2020:

Very interesting article Rupert

Miebakagh Fiberesima from Port Harcourt, Rivers State, NIGERIA. on May 29, 2020:

Rupert, the best way to stop this sickness of gambling and the loss of money is to stop gambling with one's life. Gamlers did not win anything. Whatever their won soon lost all in re-gambling on the belief that a greater is ahead. Thanks.