Why does the house always win in the long run, if every spin is a coin flip?
In an internment camp in 1940s Denmark, a mathematician flipped one coin 10,000 times to check a theorem a Swiss mathematician had needed twenty years to prove.
▶ Start the storyA casino can lose money on any single spin of the roulette wheel. But the game is not quite a fair coin flip: its payouts would let the casino break even if the wheel had only 36 numbers, and having 37 or more numbers gives the casino its edge. On a single-zero European wheel, a bet on one number loses 2.70% of the stake on average. Over a large number of spins the casino's earnings tend towards that predictable percentage, and a player's winning streak is eventually overcome by the parameters of the game. That is the law of large numbers: the average of the results from a large number of independent random trials converges to the true value. Roll a six-sided die often enough and the average of the rolls approaches 3.5, the expected value of one roll. Flip a fair coin often enough and the proportion of heads approaches one half.
The law has a catch that traps gamblers: it speaks only about large numbers. Nothing guarantees that a small number of observations will match the expected value, or that a streak will soon be balanced out by the opposite result. After four heads in a row, the next flip of a fair coin is still fifty-fifty, because the probability of a run continuing for one more toss is always one half. This mistake is called the gambler's fallacy, or the Monte Carlo fallacy, after the roulette wheel that spun black 26 times in a row at the Monte Carlo Casino in 1913.
The law has a human history. Jacob Bernoulli took over twenty years to make the proof of a special case rigorous and called it his golden theorem. It was published in 1713, eight years after his death. Then, in April 1940, the mathematician John Kerrich was caught by the Nazi invasion of Denmark while visiting his in-laws in Copenhagen, and was interned. With a fellow internee, Eric Christensen, he tossed a coin 10,000 times, recording the heads as they went, and watched the proportion approach 50 percent. The final count was 5,067 heads.
5,067
More flips cannot fix everything. If the trials embed a selection bias, the bias remains however many trials you add.
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Recap
Averages settle down with many trials, but a streak is never "owed" a correction: the coin has no memory.
💡 A trick to remember it · The crowd settles; the streak doesn't owe: the share of heads steadies while the raw gap keeps drifting.
Surprising fact · The proportion of heads converges to one half, yet the absolute gap between heads and tails almost surely grows.
Connects to
- 🔔 Why does the bell curve keep showing up, even when nothing is bell-shaped?
- ✈️ Why would armoring a bomber's most-hit spots be a mistake?
- 🌳 How can a computer learn Go by playing random games?
- 🎰 Why did gamblers lose millions betting against 26 blacks in a row?
- 🎲 How did an unfinished game of chance give birth to probability theory?
Sources (5)
No source, no claim. Every fact in this lesson (17 claims) cites at least one of these.