The gambler's fallacy and the law of large numbers: how are both true?
"I'm due for a win"
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A coin has no memory. Yet toss it enough, and heads really does creep toward half. Those two sound like they clash: if the coin keeps no tally, who's pulling the share back to half? Say a coin lands heads ten times in a row. A little voice says: surely tails is due next.
At a card table, the same thought goes: I've lost five in a row, so I'm due a win. It has a name: the gambler's fallacy. But the coin has no idea what it just did. No memory, no ledger. Next toss, it's still fifty-fifty. In 1913, a roulette wheel in Monte Carlo landed on black twenty-six times in a row. People bet on red, sure it was due, and lost a fortune.
So how does the share get back to half? Keep those ten heads, and keep tossing. Say every toss after that splits evenly, half and half. After a hundred more, heads is at 54.5 percent. After a thousand more, 50.5 percent. After ten thousand, just 50.05.
The share came back to half, yet not a single extra tails showed up. Those first ten heads are still there. It's like putting ten drops of ink into a tub of water. Nobody fishes the ink out. The more water you run, the paler it gets. The coin never pays anything back. It just dilutes that early luck in more and more tosses.
This is called the law of large numbers: the more times you try, the closer the share you count gets to the true odds. The ink is the extra heads at the start, the water is every toss after, and the colour of the water is the share you count. Someone really did this. A mathematician held in a wartime camp tossed one coin ten thousand times with a fellow prisoner.
Heads came up 5,067 times: 50.67 percent, close to half. Yet heads beat tails by a full 134. The gap in count wasn't small at all. What got diluted was the share. So the two don't clash. The gambler's fallacy is about the next toss. The law of large numbers is about the share after many tosses.
The next toss is always fifty-fifty. After many tosses, the early luck fades out of sight. And the one this law pays most reliably is the casino. Bet red at roulette and your odds are a bit under half. Every hundred you bet loses about 2.70 on average. Over enough spins, that 2.70 lands in the casino's pocket every time.
A coin lands heads ten times. On the eleventh, do you pick heads or tails?
The coin keeps no score
A coin lands heads ten times in a row, and a little voice says: surely tails is next. At a card table, it sounds like "I've lost five hands, I'm due." That hunch has a name: the gambler's fallacy.
But the coin has no idea what it just did. It has no memory and no ledger.
On the next flip, it's still
fifty-fifty.
In 1913, a roulette wheel at the Monte Carlo casino came up black 26 times in a row. Players kept betting on red because red was "due," and lost a fortune. That's why it's also called the Monte Carlo fallacy.
Ten drops of ink, a tub of water
So where does "it evens out to half" come from? Keep those first ten heads, assume every later flip splits perfectly fifty-fifty, and keep flipping:
Not a single extra tails was handed out, yet the share still drifts back to half. It's like ten drops of ink in a tub: nobody scoops the ink out, but the more water you pour in, the paler it gets.
The coin doesn't pay back.
The early luck just
gets diluted.
This is the law of large numbers: the more times you try, the closer the share you count gets to the true probability. The ink is the extra heads at the start, the water is every flip after, and the colour is the share you measure.
Someone really flipped 10,000 times
A mathematician held in an internment camp during the war flipped a coin 10,000 times with a fellow prisoner: 5,067 heads, or 50.67%, very close to half.
Yet heads beat tails
by a full 134.
The gap in count didn't shrink at all; what got diluted was the share. This is where the ink picture falls short: in real flips, later tosses also add random extra heads or tails, so the gap wanders and often grows. The total just grows faster, so the share still heads toward half. (The bars are cut off at the bottom so the gap is easy to see.)
Why a coin-flipping "champion" proves nothing is covered in the survivorship bias episode.
Each rule covers its own ground
In short: "it returns to half" is right. Thinking it gets there by "making up" for the past is the mistake.
This law is on
the casino's side
A European roulette wheel has 37 slots. A bet on red wins on only 18 of them, a bit less than half. On average, every 100 you bet loses 2.70.
Anyone can win a few rounds.
Over many bets, that 2.70
lands in the casino's pocket.
pays back.
Early luck only
gets diluted.
Quick check
A coin starts with ten heads. After 10,000 flips, heads is near half again. Why?
The coin keeps no score and never owes tails. Like ten drops of ink in a tub, the ink stays, but more water makes it paler.
