Law of large numbers
The law of large numbers is a fundamental principle in probability theory that asserts that as a random experiment is repeated numerous times, the average of the outcomes will converge towards the expected value or true predicted average of those outcomes. This concept emphasizes the necessity of conducting a large number of trials to accurately observe the underlying probabilities of different outcomes, as results from a limited number of trials may show significant short-term deviations. For example, flipping a coin a few times might yield an unbalanced outcome, such as tails appearing consecutively, which does not reflect the inherent 50-50 chance of heads or tails.
The law helps clarify common misconceptions in gambling, such as the gambler's fallacy, where individuals wrongly believe that past outcomes influence future results. In reality, each coin flip or game spin is an independent event, and the probability remains constant regardless of previous outcomes. The law of large numbers illustrates that while short-term variations can occur, over a significant number of trials, the average will stabilize around the expected probability. Understanding this concept can aid in making more informed decisions in situations involving chance and uncertainty.
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Law of large numbers
The law of large numbers is a concept in probability theory stating that the more times a random action is repeated, the closer the average of the action's numerous outcomes will get to the true predicted average of those outcomes. The important point in the law of large numbers is that only through frequent repetition of a random action can someone observe the real probability of each of the action's outcomes. This is because an action repeated only a few times can appear to result in one-sided outcomes due to short-term deviations from the real average. The law of large numbers is often used to debunk beliefs that gamblers can predict the next outcome in games of chance based on what has already occurred. According to the law, an outcome's true probability of occurring will always determine what happens.
Overview
The law of large numbers states that the outcomes in a series of random events essentially have more time to reach the true probability of the outcomes occurring if the events are repeated a large number of times. For instance, the heads and tails sides of a flipped coin each have a 50 percent chance of landing face up. If someone flips a coin five or ten times, the coin may land with tails up every time. However, this only appears to violate the coin's 50 percent probability of landing with heads up. The same coin flipped one hundred times would yield a distribution of heads and tails much closer to the predicted 50 percent average. The distribution would be even closer after one thousand flips. Five or ten flips are not enough to indicate the true probability of the coin's landing on heads or tails.
As its name suggests, the law of large numbers applies only to a large number of random event repetitions. The pattern of an event's true probability cannot be observed in the short term. Many people misunderstand this aspect of the law by subscribing to the law of averages. This is the assumption that the outcomes of future random events will even out any recent deviations from the event's true average. The law of averages might lead someone to assume that a coin that has landed with tails up ten consecutive times is "due" to land on heads next. However, each coin flip is its own random event and is not influenced by any previous flips. The law of large numbers dictates that an outcome's probability of occurring is revealed only as the number of event repetitions increases.
The "gambler's fallacy" is a particular application of the law of averages. It refers to the wishful thinking of gamblers who assume that because they are on a losing streak, it is "time" for the game to turn in their favor. They misunderstand the law of large numbers by thinking that a losing streak must end so the event's average of outcomes can even out. But the law of large numbers states only that this average evens out as the number of event repetitions increases. Deviations can still occur in the short term. Slot machines and roulette wheels have the same odds of success every time they are played. The law of large numbers always applies, but slot machines theoretically could pay out jackpots two, three, or more times consecutively.
Bibliography
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