Law of large numbers in gambling ·
Further reading — Module 03
When Luck Disguises Reality
Most people call it luck. A run of wins, a cold streak, a night where nothing lands — it feels like something is happening to you. In reality, what you are almost always watching is short-term variance: ordinary randomness behaving exactly as randomness does. Small samples are streaky, and streaks look like meaning. The law of large numbers explains why that impression fades the longer you play.
20 flips of a coin can look surprisingly far from an even 50/50 split of heads and tails.
5,000 flips almost always do.
The gap between those two truths is where misunderstanding thrives.
01 — The short run
20 coin flips rarely look even
Picture the simplest fair game: a coin toss. After just 20 flips, the results often look surprisingly uneven. Long streaks of heads or long streaks of tails. Patterns that seem meaningful. Yet every flip is still an independent 50/50 event.
20 coin flips. Result: 13 heads, 7 tails.
Longest streak: 6 heads in a row.
To many people, this feels extraordinary. In reality, it's exactly what randomness often produces.
02 — The long run tells a different story
5,000 coin flips — the swing toward convergence
After the first flip, the result is always either 100% heads or 100% tails because only one outcome exists. As more flips occur, the swing naturally contracts toward the centre.
As more flips are added, the proportion of heads and tails tends to settle closer to the underlying 50/50 probability.
The centre line represents an even 50/50 proportion of heads and tails.
After 5,000 flips, the tally is 2509 heads vs 2491 tails — a 0.4% swing toward heads. The first flip forced ±100%. The long-run answer sits within a hair of even.
At the start randomness can look dramatic. With only a handful of flips, the swing can move wildly because there are very few results. As more flips are added, the proportion of heads and tails tends to settle closer to the underlying 50/50 probability. In a fair game, the proportion of wins and losses tends toward 50/50 over the long run.
The view from behind the tables
“From behind the tables, these swings never looked unusual. One player left convinced they'd found a winning system. Another believed luck had finally turned in their favour. From the casino's side of the table, these were all part of the familiar story, all with the same familiar ending.”
Final thought
Now imagine the same 50/50 coin toss, with the same long-term convergence to an even underlying probability. But this time, every time you guess incorrectly you lose £10 — while every time you guess correctly, you win only £9. How would that work out in the long run?
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