The Gambler’s fallacy is a misconception of probability in which one believes that the probability of a future event is related to its previous events. Kahneman & Tversky (1974) describe this as a cognitive bias arising from the representativeness heuristic. In other words, we expect a random sequence to “appear random”. If flipping a coin, we would interpret H-T-H-T-T-H as more likely than H-H-H-T-T-T. Therefore, the gambler’s fallacy is a phenomenon in which we base our expected probability judgements on previous events. For example, if you role a dice and get a “4” several times in a row, you might believe that you are LESS likely to roll another “4” since you’ve already done it several times in a row, despite the fact that each roll (i.e. event in the random sequence) is independent of one another.
As a sort of counterpart to the gambler’s fallacy is the hot-hand fallacy. Here people predict the likelihood of a future event to be increased by previous/recent events. In a classic example, a basketball player is considered ‘hot’ if they continue to make a basket one after another. Therefore, people think their likelihood of continuing the streak is increased/the sequence of events is nonrandom.
Tversky, A., & Kahneman, D. (1974). Judgement under Uncertainty: Heuristics and Biases. Science, 185, 1124-31.