Gambler's Fallacy - A Problem Ill-defined

Monte Carlo

The origins of the Gambler’s Fallacy comes from a famous story about a roulette streak in the Monte Carlo casino. Basically the streak was sequential black on 26 spins of the roulette wheel. As the streak continued, more people began betting. But what is fallacious about this behavior?

The Problem

The mistake in the Gambler’s Fallacy is simply a misunderstanding of the problem. Since every spin of the wheel is independent then the past states (read R or black B) do not affect the future. In mathematical terms:

\[P\left(\bigcap_{i=1}^{n} p_i\right) = \prod_{i=1}^{n} P(p_i)\]

Basically $P$ represents the streak probability and $p_i$ represents the probability of either R or B (which each have probability of $\frac{1}{2}$). So then the question of predicting the probability of the 27th spin of the wheel is simply the same for either R or B:

\[\begin{align} \text{P(BBBBBBBBBBBBBBBBBBBBBBBBBBB)} &= \frac{1}{2^{27}} \\ \text{P(BBBBBBBBBBBBBBBBBBBBBBBBBBR)} &= \frac{1}{2^{27}} \end{align}\]

Instead what patrons of the Monte Carlo Casino were intuitively aware of is how rare getting a streak of pure B is in comparison to getting a non-pure combination (i.e. at least one R or one B in a streak of 27):

\[\begin{align} P(\neg (R_{27} \cup B_{27})) &= 1 - P(R_{27} \cup B_{27}) \\ &= 1 - \frac{2}{2^{27}} \\ &= 0.9999999850988388 \end{align}\]

But still a pure streak (all R or all B) while rare, does not change the probability of the next spin being either R or B.

Moral

When thinking about a problem it is extremely easy to misunderstand the context or setup of the problem, and instead frame the problem as a different problem. In this example, the patrons of the Monte Carlo Casino failed to grasp that the problem is NOT the probability of getting a non-pure streak (any combination other than all B or all R) but rather a simple coin toss (i.e. $P(\text{H}) = P(\text{T}) = \frac{1}{2}$).