Gambler’s Fallacy, explained.
The gambler's fallacy is the belief that an independent random process must soon compensate for a recent streak, making the opposite outcome more likely next.
Why it happens
A long-run balance does not create a short-run debt. For an independent fair coin, the mechanism generating the next toss has no memory of earlier tosses. More observations can dilute a past imbalance without requiring a corrective streak.
Independent random events do not remember the sequence before them.
Read the result
After a streak, distinguish the probability of the next toss from the probability of a whole specified sequence. Earlier outcomes are already known when you predict the next one.
A worked example
Five heads in a row
A fair coin has produced five heads. You now choose whether the next toss will be heads or tails.
The chance of tails on that next independent toss is still 50%.
Six specified heads were unlikely before the sequence began, but that does not make tails more likely after five have happened.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
The fair-coin chance of heads on the next flip remains 50% after any specified streak. The plot contrasts that with the declining chance of having seen the whole streak beforehand, (1/2)^n.
Where this idea is useful
A practical use
Avoid treating a losing run in a fair game as evidence that a win is owed.
A common misconception
“A streak proves the opposite result is due.”
Independence means previous outcomes do not change the next probability. A suspected biased coin is a different question about the mechanism.
What this explanation leaves out
- If the device is biased or conditions change, past results can be informative; independence is an assumption.
Can a streak ever be useful evidence?
Yes, if the probability is unknown or outcomes are dependent. It may inform a hypothesis about bias or changing conditions; it still does not create a compensating obligation.
Does the process remember its past, or are you supplying the memory yourself?
Associated thinkers
Further reading
Explore the original research or the teaching reference behind this experiment.