Borel-Cantelli Lemma

Let {A_n}_(n=0)^infty be a sequence of events occurring with a certain probability distribution, and let A be the event consisting of the occurrence of a finite number of events A_n for n=1, 2, .... Then the probability of an infinite number of the A_n occurring is zero if


Equivalently, in the extreme case of P(A_n)=0 for all n, the probability that none of them occurs is 1 and, in particular, the probability of A that a finite number occur is also 1.

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Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Netherlands: Reidel, pp. 435-436, 1988.

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Borel-Cantelli Lemma

Cite this as:

Weisstein, Eric W. "Borel-Cantelli Lemma." From MathWorld--A Wolfram Web Resource.

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