Law of averages

The law of averages is a lay term used to express the view that eventually, everything "evens out." For example: Two very similar people who drive similar cars in similar circumstances over a long period of time will have roughly the same number of accidents. The more children you have, the more likely you will have an equal division of boys and girls. The longer you flip a coin, the more likely the number of heads and tails will equalise.

While the general belief in the law of averages is why people gamble in Las Vegas on the belief that they will "sooner or later break even", the law of large numbers is why casinos in Las Vegas make billions of dollars.

The law of large numbers states that a large sample of a particular probabilistic event will tend to reflect the underlying probabilities. For example, after tossing a "fair coin" 1000 times, we would expect the result to be approximately 500 heads results, because this would reflect the underlying .5 chance of a heads result for any given flip.

However, it is important that while the average will move closer to the underlying probability, in absolute terms deviation from the expected value will increase. For example, after 1000 coin flips, we might see 520 heads. After 10,000 flips, we might then see 5096 heads. The average has now moved closer to the underlying .5, from .52 to .5096. However, the absolute deviation from the expected number of heads has gone up from 20 to 96.

There are common ways to misunderstand and misapply the law of large numbers:

• "If I flip this coin 1000 times, I will get 500 heads results." False. While we expect approximately 500 heads, it is not the case that we will always get exactly 500 heads results (in fact, while it is the most likely outcome, it is quite unlikely). If the coin is fair the chance of getting exactly 500 heads is about 2.52%. Similarly, getting 520 heads results is not conclusive proof that the coin's true probability of getting heads on a single flip is .52

• "I just got 5 tails in a row. My chances of getting heads must be very good now." False. Many probabilistic events are independent of one another, which means the result of one event does not in any way influence the outcome of another. Coin flips are independent events. The coin does not "remember" what it has flipped previously and self-adjust to get an overall average result. The coin is not "due" for a heads. The probability remains .5 for each individual flip. A belief in this fallacy can be devastating for amateur gamblers. The thought that "I have to win soon now, because I've been losing and it has to even out" can encourage a gambler to continue to bet more.