Law of averages

The law of averages is a lay term used to express a belief that outcomes of a random event will "even out" over a large sample.

As invoked in everyday life, the "law" usually reflects bad statistics or wishful thinking rather than any mathematical principle. While there is a real theorem that a random variable will reflect its underlying probability over a large sample, the law of averages typically assumes that unnatural short-term "balance" will occur.

Examples

 * Belief that an event is "due" to happen: For example, "The Roulette wheel has landed on red three consecutive times. The law of averages says it's due to land on black!" Of course, the wheel has no memory and its probabilities do not change according to past results. Similarly, there is no statistical basis for the belief that a losing sports team is due to win a game or that lottery numbers which haven't appeared recently are due to appear soon.


 * Belief that a sample's average must equal its expected value. For example, Daily Show host Jon Stewart joked that out of 10 Republican candidates for president, "the law of averages says one of these guys is a little Barney in the Franks." Even if 10% of the population is homosexual, there is no guarantee that one member in a group of ten must be homosexual. There is a measurable chance that all ten candidates are homosexual or that none are homosexual - the average only indicates that the occurrence of homosexuals in a large, randomly sampled population should approach 10%. Similarly, if you flip a fair coin 100 times, there is only an 8% chance that there will be exactly 50 heads.


 * Belief that a rare occurrence will happen given enough time: For example, "If I send my résumé to enough places, the law of averages says that someone will eventually hire me." This may actually be true assuming nonzero probabilities and the law of averages is simply named in place of the Law of Large Numbers.

Mathematics
Let $$ p $$ denote the probability of a given outcome in a single (Bernoulli) trial and $$ S_n $$ the number of occurrences of this outcome in $$ n $$ trials. Then for any number $$ \epsilon >0 $$, we have that

$$\lim_{n\to\infty}\Pr\left(\left|\frac{S_n}{n}-p\right|>\epsilon\right)=0$$

For example: If you were to toss a fair coin and keep track of the proportion of tosses that land on heads, this proportion would approach $$1/2 $$ as you toss the coin more times.

This interpretation of the Law of Averages is a special case of the Weak Law of Large Numbers as applied to Bernoulli Trials.