the gambler's fallacy
The gambler's fallacy is the belief that a run of one outcome makes the opposite outcome 'due'. After the roulette wheel lands on red five times in a row, it feels like black must be overdue and more likely next. After a string of tails, the coin seems to owe you some heads. It does not. The wheel and the coin have no memory.
The error is forgetting that the trials are independent. For independent events, P(next is heads given any past sequence) = P(heads) = 1/2, full stop — conditioning on the past changes nothing, because that is exactly what independence means. The fallacy quietly assumes a kind of self-correction that simply is not in the model. There is a real theorem nearby that gets misremembered: the law of large numbers says the long-run PROPORTION of heads tends to 1/2. But it does this by swamping early imbalances with an ever-growing pile of future tosses, never by the coin steering itself to repay a deficit. A streak is not 'corrected'; it is diluted.
There is a mirror-image mistake called the hot-hand fallacy — believing a run will continue because the process is 'on a roll'. Both come from reading patterns into genuine independence. The honest summary: with truly independent trials, the past gives zero information about the next outcome, and any feeling that an outcome is 'due' (or 'hot') is the human pattern-seeker talking, not the mathematics.
A fair coin shows TTTTT (five tails). The chance the sixth toss is heads is still exactly 1/2 — the coin does not 'remember' the tails. What was unlikely beforehand was the whole run TTTTTT (1/64), but given five tails already happened, the last toss is an even bet.
Independence means the next toss ignores the streak; only the full sequence was ever rare.
The law of large numbers does NOT make outcomes 'even out' to repay a streak — it dilutes early imbalances with sheer volume of future trials. Streaks are diluted, never corrected.