Editing Law of averages (section)

==Examples==

===Gambler's fallacy===
The [[gambler's fallacy]] is a particular misapplication 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.<ref>{{Cite news|url=https://www.forbes.com/sites/davidschwartz/2018/06/04/how-casinos-use-math-to-make-money-when-you-play-the-slots/#38cfaa0394d0|title=How Casinos Use Math To Make Money When You Play The Slots|last=Schwartz|first=David G.|work=Forbes|access-date=2018-09-12|language=en}}</ref>

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 is guaranteed (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. See [[parimutuel betting]].)

===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 the single most likely outcome, there is only an 8% chance of it occurring according to <math>P(X = 50 \mid n = 100, p = 0.5)</math> of the [[binomial distribution]]. Predictions based on the law of averages are even less useful if the sample [[bias (statistics)|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 [[asymptote|approaches but never quite reaches]] 100%.

===Chicago Cubs===
The [[Steve Goodman]] song "[[A Dying Cub Fan's Last Request]]" mentions the law of averages in reference to the [[Chicago Cubs]] lack of championship success. At the time Goodman recorded the song in 1981, the Cubs had not won a [[National League (baseball)|National League]] championship since [[1945 Chicago Cubs season|1945]], and had not won a [[World Series]] since [[1908 World Series|1908]]. This futility would continue until the Cubs would finally win both in [[2016 Chicago Cubs season|2016]].