Gambler's Fallacy
Grant Sterling
On August 18, 1913, at the casino in Monte Carlo, black came up on the roulette wheel a record twenty-six times in succession. There was a near-panicky rush to bet on red, beginning about the time black had come up a phenomenal fifteen times.
Players doubled and tripled their stakes, believing that after black came up the twentieth time there was not a chance in a million of another repeat. In the end, the unusual run enriched the casino by some millions of francs.Darrell Huff and Irving Geis, How to Take a Chance
The gambler’s fallacy (GF) is committed in the context of random, unconnected events. When (by chance) a certain outcome occurs very often in one period of time, the fallacious reasoner assumes that the opposite outcome will be more likely to occur in the future to “even out” the results.
Observer: That guy just flipped a coin 10 times and it came up “heads” every time.
Gambler: Let’s go bet on “tails” for the next flip!
As with most fallacies, GF is prevalent because it is similar to a kind of good reasoning. It is true that over a very long period of trials luck tends to “even out.” If you flip a coin a billion times, you’re likely to end up with
approximately the same number of “heads” and “tails.” But that’s only because a series of a billion random flips randomly produces even results - there’s no mechanism by which the coin “keeps track” of how many heads and tails have gone before and adjusts itself to balance things out.
Consider the following case. Suppose I intend to flip a coin 20 times, and I ask you to bet on how many times it will come up “heads.” Assuming a fair coin, you should guess 10. Now suppose I start flipping and get six “heads” in a row. I offer to allow you to change your bet. With 14 flips remaining, you should assume that “heads” will come up (approximately) half the time on the remaining flips.
So you should change your bet to 13 (six already flipped plus seven more). Luck is “evening out” in the sense that the percentages of “heads” will be expected to go down from 100% (the current results) to 65%. But there’s no reason to think that the next flip is more than 50% likely to come up “tails” or that a streak of “tails” is likely. If I were flipping 100 times, you should bet on 53 “heads” (six already flipped plus 47 [50% of the remaining 94 flips] more). So now the expected outcome is 53%. After a billion flips, you should expect 6+ 499,999,997 [50% of the remaining 999,999,994 flips]=500,000,003 total “heads,” which rounds to 50%. The more flips you perform, the closer the total percentage can be expected to come to 50%. In that sense, luck evens out. But there is no reason to think that even with a billion flips, there will be any “extra” flips that result in “tails” to even out the original streak of six.GT is also called, among other things, the Monte Carlo fallacy (from the incident quoted above) or fallacy of the maturity of chances. It’s converse is sometimes called the hot hand fallacy, in which the opposite reasoning is used: the arguer assumes that because a string of results have turned up one way, this “streak” will continue - in the coin example above, the reasoner would assume that “heads” would come up again since “heads” are “on a roll.” This sort of reasoning is at least slightly less fallacious, since there’s a chance (however small) that a streak indicates that the outcomes are not actually random (maybe the coin is actually weighted or the roulette wheel is out of balance).
Notice that GF covers only situations where the results are based on chance. In a game of skill, for example, a streak may indicate the superiority of one player or team (and hence that the streak is likely to continue!) or it may cause the other players to adjust their strategy to stop the currently successful run.
The fallacy also doesn’t cover situations where the outcome of one trial affects the outcome of later trials. If I am drawing cards from a deck and not replacing them, if the first two cards I draw are aces, it would be rational to bet against another ace being drawn next, since the previously drawn aces are no longer in the deck.
The easiest way to avoid this fallacy is not to think that heads-or-tails kinds of occurrences will more or less likely yield heads or tails based upon a past frequency - two, four, 40, or 400 heads does not mean that tails will come up next, and vice versa. Doing some research on statistics will help too.