The Monte Carlo Fallacy – A Gambling PSA

Note: this post assumes some knowledge of basic probability notation.

Mathematics tends to get to the truth of a situation, as pure Mathematics is never wrong. This is a result of the fact that once something is proven, it is proven forever. If Mathematics has something to say about a certain situation, you would do well to listen to the said Mathematics. As again, pure Mathematics is never wrong.

The first recorded instances of gambling date back to China circa 2300 BC, but it has probably existed for as long as humans have communicated with one another. Needless to say, gambling is a large part of the human experience.  Games of chance tend to be simple, somewhat abstract with clearly defined rules. Three attributes that have attracted Mathematicians across the ages to study these games of chance. So when it comes to gambling, there are some Mathematical results you would do well to keep in mind.

Probability has actually been studied using the axiomatic approach. Starting with three axioms, known as the Kolmogorov axioms many interesting results have been deduced. One concept that has been formally defined is the idea of statistical independence, which put simply is the idea of two events that don’t affect each other. The formal definition states that:

“Two events A & B are independent if and only if their joint probability equals the product of their probabilities”

That is, P(A and B both occur)=P(A occurs) * P(B occurs). This may not seem like the most intuitive definition of this concept, but rewriting it as conditional probability illustrates it is correct more clearly.


This shows that if two events are independent then the probability of A occurring is the same as the probability of A occurring given B has already occurred. I.e. What happens with the event B has no relation on A. It seems simple, if a little abstract, when discussed like this. However this simple idea is in fact a widely spread fallacy known as the Gambler’s fallacy.

To illustrate the Gambler’s fallacy, let’s take the example of a coin flip. I’m very aware it is perhaps the most overused and dull example scenario in all of mathematics, but it does the job nonetheless. When flipping a coin in succession each flip is independent from the last one, the coin does not remember what has come up, and so the coin flips follow the rules given above. However in the case where say, tails has come up for the last few flips, many of us think it’s natural that heads is ‘due’ and more likely to come up next. This is the Gambler’s fallacy.

Note: It is the ratio between heads and tails that tends towards a half over a large number of tosses, the difference between the number of tosses does not tend to zero. This is as a result of the law of large numbers.

Simulation of coin tosses (wikipedia)

One notable example of the Gambler’s fallacy in action happened at the Monte Carlo Casino on August 18, 1913. In a game of roulette, the ball landed on black 26 times in a row. This led to many punters betting heavily on red as they believed that the roulette wheel was now due a long streak of red, but as roulette spins are independent from each other this was not the case. The casino made millions of francs from these bets.

So next time you find yourself gambling, take note of the gambler’s fallacy and try to remember that each new coin flip, dice roll or roulette spin is a new fresh event and whatever has come before has zero effect.

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