Law of truly large numbers

The law of truly large numbers is the observation in statistics that any highly unlikely result is likely to occur, given a large enough number of independent samples. It is not a mathematical law, but a colloquialism. The law has been used to rebut pseudo-scientific claims.
The observation is attributed to statisticians Persi Diaconis and Frederick Mosteller. Skeptic and magician Penn Jillette similarly said that "million-to-one odds happen eight times a day" among the roughly 8 million inhabitants of New York City. In another illustrative class of cases—which also involve combinatorics—lottery drawing numbers have been duplicated in close or even immediate succession.

Examples

Suppose that an event has only a 1% probability of occurring in a single trial. Then, within a single trial, there is a 99% probability that will not occur. However, if 100 independent trials are performed, the probability that does not occur in a single of them, even once, is. Therefore, probability of occurring in at least one of 100 trials is . If the number of trials is increased to 1,000, that probability rises to. In other words, a highly unlikely event, given enough independent trials, is very likely to occur.
Similarly, for an event with "one in a billion odds" of occurring in any single trial, across 1 billion independent trials the probability of occurring at least once is. Taking a "truly large" number of independent trials like 8 billion raises this to .
These calculations can be formalized in mathematical language as: "the probability of an unlikely event X happening in N independent trials can become arbitrarily near to 1, no matter how small the probability of the event X in one single trial is, provided that N is truly large."
For example, where the probability of unlikely event X is not a small constant but decreased in function of N, see graph.
In high availability systems even very unlikely events have to be taken into consideration, in series systems even when the probability of failure for single element is very low after connecting them in large numbers probability of whole system failure raises.

In criticism of pseudoscience

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 outnumber 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.