Streaks Are Normal: The Law of Large Numbers
Flip a fair coin 100 times and you will probably see something that feels broken: a run of six, seven, maybe eight identical results in a row. Nothing is broken. The law of large numbers — the theorem that averages converge to expectations as trials pile up — says nothing at all about your next ten flips. Randomness is wild up close and orderly only from a distance, and confusing those two views is the source of half the superstition in games.
What the law actually promises
Toss a coin forever and the proportion of heads approaches 50%. That's all. It does not promise heads and tails alternate politely, that deficits get corrected (that's the gambler's fallacy), or that 1,000 flips lands on exactly 500. In fact the expected absolute gap between heads and tails counts grows as trials increase — around 25 after 1,000 flips — even while the percentage gap shrinks toward zero. The ratio converges; the raw difference wanders. Both statements are true, and holding them simultaneously is most of the wisdom.
Streaks, quantified
In n fair flips, the longest expected run of identical results is roughly log₂(n). One hundred flips: expect a longest streak around 6-7. A thousand: around 10. When a basketball fan sees a player hit seven straight, or a Catan player watches the 6 hit four turns running, the honest response is rarely "something's up" — it's "n is large." Streaks aren't violations of randomness. They're its signature. Sequences without streaks are the suspicious ones, which is exactly how statisticians catch humans faking coin-flip data: fake sequences alternate too much.
How casinos eat
The law of large numbers is the entire business model of gambling. American roulette gives the house a 5.26% edge — meaning any given night, any given player can win big (short run: wild). But the casino isn't playing your session; it's playing millions of spins a year (long run: orderly), where revenue converges to edge × volume with industrial reliability. Casinos don't beat gamblers. Arithmetic does, at scale. Insurance companies run the identical play with premiums: individual claims are chaos, a million policies are a spreadsheet.
Generate your own streaks
Flip 50 times and watch the streak counter — runs of five-plus appear like clockwork, exactly as the math predicts.
Test it live →Living with both truths
Short-run wildness means: don't judge a strategy, a player or a wheel by ten trials; expect clusters; treat small samples as gossip, not evidence. Long-run order means: edges compound relentlessly, so tiny advantages (and tiny fees) matter enormously at scale; and any system profiting from volume — casino, insurer, index fund — is riding the same theorem. Respect the short run's chaos, trust the long run's arithmetic, and never confuse which one you're standing in.