Gambler's Fallacy - Non-examples of The Fallacy

Non-examples of The Fallacy

There are many scenarios where the gambler's fallacy might superficially seem to apply, but actually does not. When the probability of different events is not independent, the probability of future events can change based on the outcome of past events (see statistical permutation). Formally, the system is said to have memory. An example of this is cards drawn without replacement. For example, if an ace is drawn from a deck and not reinserted, the next draw is less likely to be an ace and more likely to be of another rank. The odds for drawing another ace, assuming that it was the first card drawn and that there are no jokers, have decreased from 4⁄₅₂ (7.69%) to 3⁄₅₁ (5.88%), while the odds for each other rank have increased from 4⁄₅₂ (7.69%) to 4⁄₅₁ (7.84%). This type of effect is what allows card counting schemes to work (for example in the game of blackjack).

Meanwhile, the reversed gambler's fallacy may appear to apply in the story of Joseph Jagger, who hired clerks to record the results of roulette wheels in Monte Carlo. He discovered that one wheel favored nine numbers and won large sums of money until the casino started rebalancing the roulette wheels daily. In this situation, the observation of the wheel's behavior provided information about the physical properties of the wheel rather than its "probability" in some abstract sense, a concept which is the basis of both the gambler's fallacy and its reversal. Even a biased wheel's past results will not affect future results, but the results can provide information about what sort of results the wheel tends to produce. However, if it is known for certain that the wheel is completely fair, then past results provide no information about future ones.

The outcome of future events can be affected if external factors are allowed to change the probability of the events (e.g., changes in the rules of a game affecting a sports team's performance levels). Additionally, an inexperienced player's success may decrease after opposing teams discover his weaknesses and exploit them. The player must then attempt to compensate and randomize his strategy. (See Game theory).

Many riddles trick the reader into believing that they are an example of the gambler's fallacy, such as the Monty Hall problem.