The law of truly large numbers, attributed to Persi Diaconis and Frederick Mosteller, states that with a large enough number of samples, any outrageous thing is likely to be observed. Because we never find it notable when likely events occur, we highlight unlikely events and notice them more. The law is often used to falsify different pseudo-scientific claims, as such it and its use is sometimes criticized by fringe scientists. The law is meant to make a statement about probabilities and statistical significance: in large enough masses of statistical data, even minuscule fluctuations attain statistical significance. Thus in truly large numbers of observations, it is paradoxically easy to find significant correlations, in large numbers, which still do not lead to causal theories, and which by their collective number, might lead to obfuscation as well. The law can be rephrased as "large numbers also deceive", something which is counter-intuitive to a descriptive statistician. More concretely, skeptic Penn Jillette has said, "Million-to-one odds happen eight times a day in New York".
Example
For a simplified example of the law, assume that a given event happens with a probability for its occurrence of 0.1%, within a single trial. Then, the probability that this so-called unlikely event does not happen in a single trial is 99.9%. Already for a sample of 1000 independent trials, however, the probability that the event does not happen in any of them, even once, is only 0.9991000 ≈ 0.3677 = 36.77%. Then, the probability that the event does happen, at least once, in 1000 trials is or 63.23%. This means that this "unlikely event" has a probability of 63.23% of happening if 1000 independent trials are conducted, or over 99.9% for 10,000 trials. The probability that it happens at least once in 10,000 trials is In other words, a highly unlikely event, given enough trials with some fixed number of draws per trial, is even more likely to occur. This calculation can be generalized, formalized to use in straightforward mathematical proof that: "the probability c for the less likely event X to happen in N independent trials can become arbitrarily near to 1, no matter how small the probability a of the event X in one single trial is, provided that N is truly large."
The law comes up in criticism of pseudoscience and is sometimes called the Jeane Dixon effect. It holds that the more predictions a psychic makes, the better the odds that one of them will "hit". Thus, if one comes true, the psychic expects us to forget the vast majority that did not happen. Humans can be susceptible to this fallacy. Another similar manifestation of the law can be found in gambling, where gamblers tend to remember their wins and forget their losses, even if the latter far outnumbers the former. Mikal Aasved links it with "selective memory bias", allowing gamblers to mentally distance themselves from the consequences of their gambling by holding an inflated view of their real winnings.
