Further reading — Module 03
Short term variance
vs long term convergence
20 flips of a coin rarely come close to 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.
The centre line represents perfect balance: an equal number 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 over time, the variance shrinks and results converge toward the expected value. In a fair game, that expected value is zero — wins and losses balance out. No one wins, no one loses.
Final thought
Now imagine the same 50/50 coin toss, with the same long-term convergence to zero expected value. But this time, every time you guess correctly you win £10 — and every time you don't, you lose £11. How would that work out long run?
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Further reading in Module 03