The Borel-Cantelli lemma guarantees with probability 1 that only finitely many events
in a sequence occur when the sum
of their probabilities converges.
Let 
be a sequence of events occurring
with a certain probability distribution,
and let 
be the event consisting of the occurrence of a finite
number of events 
for 
, 2, .... Then the probability
of an infinite number of the 
occurring is zero if
 |
As a special case, if 
for all 
, the probability that none
of them occurs is 1 and, in particular, the probability
of 
that a finite number occur is also 1.
