Ask Uncle Colin: A Gambling Fallacy

Dear Uncle Colin,

Does the Law Of Large Numbers contradict the fact that the Gambler’s Fallacy is a fallacy?

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Hi, GAMBLING, and thanks for your message!

The short answer is: no. I imagine you want a bit more detail than that, though! Let’s start, for those unfamiliar, with definitions of the two things.

The Law Of Large Numbers

The Law Of Large Numbers states that, if you repeatedly conduct independent Bernoulli trials ¹ with probability $p$, then - over the long term - the proportion of successful trials approaches $p$.

The Gambler’s Fallacy

The Gambler’s Fallacy is the incorrect belief that statistics has a memory - and that, for instance, if you haven’t rolled a six on a die for some time, a six is somehow more likely to come up that it normally would be.

Do you see the contradiction?

How can the Law Of Large Numbers (proportion tends towards underlying probability) be correct, without the Gambler’s Fallacy (the universe evens things up)?

It comes down to something slightly subtle: the Law Of Large Numbers says nothing about the number of successes tending to the expected value, only the proportion.

In particular, if you perform $n$ independent Bernoulli trials, each with probability $p$, the number of successes is drawn from a binomial distribution, $X\sim B(n,p)$.

The expected value of $X$ is $np$, with a standard deviation of $\sqrt{np(1-p)}$ - which gets larger as $n$ does. The number of successes, on average, gets further from the expected value as $n$ increases.

However, the expected proportion of successes is $p$, with a standard deviation of $\frac{\sqrt{p(1-p)}}{\sqrt{n}}$. This standard deviation gets smaller as $n$ increases - so the proportion of successes approaches $p$ even though there’s nothing to move the number of successes towards $np$.

Hope that helps!

- Uncle Colin

* Edited 2017-12-28 after Adam pointed out that correct things do indeed contradict fallacies.

Footnotes:

1. Bernoulli trials are just events which either succeed or fail with a fixed probability