The Gambler’s Fallacy

A fair coin is flipped ten times and results in eight heads and two tails. Does this mean that more tails will show up in the next few flips? Those who subscribe to this fallacy believe that the “law of averages” will rescue them in the midst of a losing streak by turning things around.

After all, in the long run, there should be 50% heads and 50% tails, so more tails should be coming up now to “even out” things. Others might expect more heads to show up because heads are “hot.” Both views are incorrect.

Because the tosses are independent trials, the probability of heads is 50% on each and every toss regardless of what happened on previous tosses.

Situations similar to this coin example occur in the casino. Suppose, for example, that eight spins of the roulette wheel result in eight consecutive red numbers. Is black now more likely on the ninth spin because it is “due,” or is red more likely now because it is hot?

Many people believe that if one color has come up several times in a row in roulette then the other color is now either more or less likely. The truth, of course, is that each color has the same probability of occurring on every spin, regardless of the outcomes of the past spins.

Viewing it from the long term, for every sequence of eight consecutive reds that is followed by a black on the ninth spin, there will another sequence of eight reds followed by a ninth red. The bottom line is that in a game like roulette, every trial is independent and past history is irrelevant.

This principle applies to other casino games involving independent events. The dice may appear to be “hot” at a craps table, but this does not change the odds of the shooter’s throwing the various numbers. Similarly, keno numbers do not turn hot or cold. For independent events, each outcome is entirely unrelated to past outcomes and so “hot” and “cold” streaks do not really exist in the sense that particular outcomes become more or less likely to occur than normal.

Streaks are just statistical fluctuations and no one can predict whether a streak will continue or when it will end.

Some betting systems use the fallacious logic of riding streaks. Others use the equally faulty logic of betting against the streak.

The former, of course, is based on the hot streak fallacy; the latter on the mistaken notion that in the short run things must even out according to the “law of averages.”

Considered thought about the nature of independent events and the correct application of the law of averages will reveal these systems to be worthless.