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+----- Thread: Gambler's fallacy (/showthread.php?tid=1797)
Gambler's fallacy - mklabgr - 09-02-2026
Summary
The gambler’s fallacy is the mistaken belief that past outcomes of independent random events influence what will happen next. For example, after a fair coin lands heads several times in a row, someone may believe that tails is now “due.” In reality, each toss is independent, so the probability remains
$P(heads)=P(tails)=12.P(\text{heads})=P(\text{tails})=\frac12$.
Likewise, if four heads have already occurred, the probability that the fifth toss is heads is still $1/2$. The confusion comes from mixing up the probability of predicting an entire sequence beforehand with the probability of the next event after part of the sequence has already occurred. Five consecutive heads have probability $1/2^5=1/32$ before the experiment begins, but once four heads have already occurred, the probability of another head is simply $1/2$.
The fallacy is also called the Monte Carlo fallacy, after a famous roulette event at the Monte Carlo Casino on August 18, 1913. The ball reportedly landed on black 26 times consecutively, leading gamblers to bet increasingly large sums on red because they believed red had to appear soon. But if the roulette wheel was unbiased, each new spin remained independent of the preceding spins. The psychological explanation is closely associated with what Kahneman and Tversky called the representativeness heuristic: people expect even short random sequences to look balanced, although genuine randomness naturally produces streaks and clusters.
Importantly, the gambler’s fallacy applies only when events really are independent and probabilities remain unchanged. Drawing cards without replacement is different: after an ace is removed from a deck, the probability of drawing another ace genuinely changes. Similarly, an extraordinary run of identical results might provide evidence that a coin, roulette wheel, or other mechanism is biased; in that case, updating probabilities based on the observations can be rational rather than fallacious.
Key takeaways
- Independent random events do not remember previous outcomes.
- A long streak does not make the opposite outcome “due.”
- Random sequences naturally contain surprisingly long streaks.
- The law of large numbers does not require short-term imbalances to be immediately corrected.
- Past outcomes matter when events are dependent, probabilities change, or observations provide evidence that the underlying process is biased.
ARTICLE