Explanation

The gambler’s fallacy is the fallacy of assuming that short-term deviations from probability will be corrected in the short-term. Faced with a series of events that are statistically unlikely, say, a serious of 9 coin tosses that have landed heads-up, it is very tempting to expect the next coin toss to land tails-up. The past series of results, though, has no effect on the probability of the various possible outcomes of the next coin toss.

Example

(1) This coin has landed heads-up nine times in a row.
Therefore:
(2) It will probably land tails-up next time it is tossed.

This inference is an example of the gambler’s fallacy. When a fair coin is tossed, the probability of it landing heads-up is 50%, and the probability of it landing tails-up is 50%. These probabilities are unaffected by the results of previous tosses.

The gambler’s fallacy appears to be a reasonable way of thinking because we know that a coin tossed ten times is very unlikely to land heads-up every time. If we observe a tossed coin landing heads-up nine times in a row we therefore infer that the unlikely sequence will not be continued, that next time the coin will land tails-up.

In fact, though, the probability of the coin landing heads-up on the tenth toss is exactly the same as it was on the first toss. Past results don’t bear on what will happen next.