Law of averages

# Law of averages

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Encyclopedia
The law of averages is a lay term used to express a belief that outcomes of a random event will "even out" within a small 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 very large sample
Law of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...

, the law of averages typically assumes that unnatural short-term "balance" must occur. Typical applications of the law also generally assume no bias
Skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. The skewness value can be positive or negative, or even undefined...

in the underlying probability distribution, which is frequently at odds with the empirical evidence
Empirical research
Empirical research is a way of gaining knowledge by means of direct and indirect observation or experience. Empirical evidence can be analyzed quantitatively or qualitatively...

.

## Examples

• Belief that an event is "due" to happen: For example, "The roulette
Roulette
Roulette is a casino game named after a French diminutive for little wheel. In the game, players may choose to place bets on either a single number or a range of numbers, the colors red or black, or whether the number is odd or even....

wheel has landed on red in three consecutive spins. 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. So even if the wheel has landed on red in ten consecutive spins the probability that the next spin will be black is still 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"). (In fact, if the wheel has landed on red in ten consecutive spins, that is strong evidence that the wheel is not fair - that it is biased toward red. Thus, the wise course on the eleventh spin would be to bet on red, not on black: exactly the opposite of the layman's analysis.) 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. This sort of belief is called the gambler's fallacy
Gambler's fallacy
The Gambler's fallacy, also known as the Monte Carlo fallacy , and also referred to as the fallacy of the maturity of chances, is the belief that if deviations from expected behaviour are observed in repeated independent trials of some random process, future deviations in the opposite direction are...

.

• Belief that a sample's average must equal its expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

. For example, if one flips a fair coin
Fair coin
In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair 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 that the number of trials is really large enough; the law of averages is then simply the Law of large numbers
Law of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...

.