The Gambler’s Fallacy

by | Oct 26, 2017


On August 18th, 1913  something unusual occurred at a roulette table at the Monte Carlo Casino. At a particular table the ball had fallen on black many times in a row. After each successive black result gamblers placed more and more money on red thinking that after so many blacks in a row that red was “due.” The ball finally fell on red on the 27th spin. Millions of dollars had been lost by that point by gamblers betting on red who had suffered from a common mathematical misunderstanding.

The chance of a European roulette wheel coming up black 26 times in a row is 1 in 145 million. Thus, it would appear that each spin of another black was increasingly less likely as the chances of 27 blacks in a row is 1 in 298 million. Of course, this is not the case – each spin is independent of every other spin and the chance of a black vs. a red is nearly 50/50 (a European wheel has 37 slots, 18 red, 18 black and zero). Thus, the chance of a Red on the first spin is 1 in 2.06, the same as it is on the 10th spin and the 27th spin.

Here’s the common misunderstanding- if we know that a series of random/independent occurrences should have an expected average value, we consciously or unconsciously expect that future occurrences will act to “force” the series of observations back to their expected average.

For example, assume a fair coin is flipped 20 times with 16 heads and 4 tails.  What is the expectation for the next 20 flips? Should we expect more tails to pull the average number of heads/tails closer to 50/50?  That may be our gut instinct, but that is wrong.  The coin has no memory of the prior 20 flips and there is no force that will cause a greater number of tails.

Instead, the law of large numbers says that as  number of trials of a random process increases, the percentage difference between the expected and actual values goes to zero. In other words, as you have more and more observations of a fair coin, the actual experience of heads and tails will approach your expectation of 50/50. This is a subtle difference – there is no “force” causing more tails after a string of heads – it’s just random variability.

Imagine flipping a coin one million times. Actual observations  of heads/tails will be very close to 50/50 because one million observations is a large number. However, it is not unusual to think that within that million coin flips there could be a string of 10 heads in a row, or 16 heads out of 20 flips, etc.  All sort of random patterns will occur within that string of one million flips. Our problem is that our experience with coin flips or roulette spins or most things in life is that we deal with very small sample sizes which may contain great variability.


  1. The human brain is really bad at handling true randomness. We find strings of similar occurrences difficult to accept even though over a large number of iterations they will invariably occur. When Apple came out with the original iPod and the shuffle feature people would freak when they got 3 Springsteen songs in a row out of their entire music collection. Apple was accused of favoring music that it may have some financial interest in. The reality was/is that sometimes you do just get 3 Springsteen songs in a row. Interestingly, one of the algorithms used by the IRS to detect fraudulent tax returns looks for returns that do not use consecutive or round numbers. When making up numbers tax cheats will invariably use non-sequential seemingly random numbers. The reality is that you should have some sequential and round numbers somewhere in there. This bias is really hard to fight. I’ve lost a lot of money at the Black Jack table thinking there is no way she can deal me another 14.

  2. I guess Momma was right, “Life IS like a box of chocolates.”

  3. If the sample size was 1 million spins of the roulette wheel would the odds migrate closer to 1 in 2 rather that the stated odds of 1 in 2.06?

    • The long term expectation for a European Roulette wheel is that red will be 18/37, black 18/37 and zero/blank 1/37. 18/37 is 48.64% or 1 in 2.06.

      • Makes sense…I dont play roulette and forgot that there was a zero slot on the wheel.

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