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Gambling pushes people to the brink of hope, yet even after repeated losses, many gamblers cling to the belief that a lucky streak is imminent. They point to a run of black outcomes on the roulette wheel and feel the next spin must be red, citing the so‑called “law of averages.” In reality, this is a misconception.
The phrase “law of averages” isn’t a formal scientific principle like Ohm’s or Newton’s laws. It stems from a common misunderstanding of the law of large numbers, a theorem first formalised in the 1700s by Swiss mathematician Jakob Bernoulli while studying probability in games such as roulette and craps.
According to the law of large numbers, as the number of independent trials grows, the observed frequencies converge toward the true probabilities. However, convergence only becomes apparent after a very large number of repetitions—often in the thousands. Assuming that this principle guarantees balance within a single gaming session is the essence of the gambler’s fallacy.
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A gambler who falls into this trap might think that after ten consecutive reds on a roulette wheel, black is “due” on the next spin. This belief is fallacious because each spin is an independent event with the same probability as every other spin. Past outcomes exert no influence on future results.
While mathematical models can calculate the probability of a specific outcome—such as rolling a six on a fair die—those calculations cannot be altered by the number of rolls already performed. The law of large numbers simply tells us that the proportion of outcomes will approach the expected distribution only after many trials, not after a handful.
Consider a coin toss, the simplest example with only two outcomes. No matter how many times you flip, you cannot guarantee an even number of heads and tails with 100 % certainty. Even achieving a 95 % chance of balance would require over 1,000 flips. This illustrates how intuitive expectations of “balance” are misleading when applied to a few spins.
