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Borel's law of large numbers, named after Émile Borel, states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event is expected to occur approximately equals the probability of the event's occurrence on any particular trial; the larger the ...
The law of truly large numbers (a statistical adage), attributed to Persi Diaconis and Frederick Mosteller, states that with a large enough number of independent samples, any highly implausible (i.e. unlikely in any single sample, but with constant probability strictly greater than 0 in any sample) result is likely to be observed. [1]
This is an accepted version of this page This is the latest accepted revision, reviewed on 17 January 2025. Observation that in many real-life datasets, the leading digit is likely to be small For the unrelated adage, see Benford's law of controversy. The distribution of first digits, according to Benford's law. Each bar represents a digit, and the height of the bar is the percentage of ...
Usually, this estimator is the proportion of times that the number occurs in the data set. If the points in the plot tend to converge to a straight line for large numbers in the x axis, then the researcher concludes that the distribution has a power-law tail. Examples of the application of these types of plot have been published. [61]
For example, in the game of bridge, the probability that a player will be dealt 13 cards of the same suit is extremely low (Littlewood calculates it as ). While such a deal might seem miraculous, if one estimates that 2 ⋅ 10 6 {\displaystyle 2\cdot 10^{6}} people in England each play an average of 30 bridge hands a week, it becomes quite ...
An essential characteristic of the typical set is that, if one draws a large number n of independent random samples from the distribution X, the resulting sequence (x 1, x 2, ..., x n) is very likely to be a member of the typical set, even though the typical set comprises only a small fraction of all the possible sequences.
Insensitivity to sample size is a cognitive bias that occurs when people judge the probability of obtaining a sample statistic without respect to the sample size.For example, in one study, subjects assigned the same probability to the likelihood of obtaining a mean height of above six feet [183 cm] in samples of 10, 100, and 1,000 men.
The lemma is often used in the proofs of theorems concerning sums of independent random variables such as the strong Law of large numbers. The lemma is named after the German mathematician Leopold Kronecker .