Law of averages

The law of averages is a layman's term for a belief that the statistical distribution of outcomes among members of a small sample must reflect the distribution of outcomes across the population as a whole.

As invoked in everyday life, the "law" usually reflects wishful thinking or a poor understanding of statistics rather than any mathematical principle. While there is a real theorem that a random variable will reflect its underlying probability over a very large sample, the law of averages typically assumes that unnatural short-term "balance" must occur.^[1] Typical applications of the law also generally assume no bias in the underlying probability distribution, which is frequently at odds with the empirical evidence.

Examples

Gambler's fallacy

The gambler's fallacy is a particular application of the law of averages in which the gambler believes that a particular outcome is more likely because it has not happened recently, or (conversely) that because a particular outcome has recently occurred, it will be less likely in the immediate future.

As an example, consider a roulette wheel that has landed on red in three consecutive spins. An onlooker might apply the law of averages to conclude that on its next spin it must (or at least is much more likely to) land on black. Of course, the wheel has no memory and its probabilities do not change according to past results. So even if the wheel has landed on red in ten or a hundred consecutive spins, the probability that the next spin will be black is still no more than 48.6% (assuming a fair European wheel with only one green zero; it would be exactly 50% if there were no green zero and the wheel were fair, and 47.4% for a fair American wheel with one green "0" and one green "00"). Similarly, there is no statistical basis for the belief that lottery numbers which haven't appeared recently are due to appear soon. (There is some value in choosing lottery numbers that are, in general, less popular than others — not because they are any more or less likely to come up, but because the largest prizes are usually shared among all of the people who chose the winning numbers. The unpopular numbers are just as likely to come up as the popular numbers are, and in the event of a big win, one would likely have to share it with fewer other people.)

On the other hand, modern slot machines are usually rigged so they do give wins a certain proportion of the time — the results are not truly random. This is carefully managed so as to encourage people to keep playing, while the casino takes its designated amount of profit.

Expectation values

Another application of the law of averages is a belief that a sample's behaviour must line up with the expected value based on population statistics. For example, suppose a fair coin is flipped 100 times. Using the law of averages, one might predict that there will be 50 heads and 50 tails. While this is admittedly the single most likely outcome, there is only an 8% chance of it occurring. Predictions based on the law of averages are even less useful if the sample does not reflect the population.

Repetition of trials

In this example, one tries to increase the probability of a rare event occurring at least once by carrying out more trials. For example, a job seeker might argue, "If I send my résumé to enough places, the law of averages says that someone will eventually hire me." Assuming a non-zero probability, it is true that conducting more trials increases the overall likelihood of the desired outcome. However, there is no particular number of trials that guarantees that outcome; rather, the probability that it will already have occurred approaches but never quite reaches unity.

See also

• Law of large numbers
• Gambler's fallacy

References