Mathematics
The Borel-Cantelli Lemma and Almost Sure Convergence
Quick fact
The Borel-Cantelli lemma states that if the sum of probabilities of a sequence of events is finite, then the probability that infinitely many of them occur is zero. This means that in an infinite sequence of independent trials, any event with positive constant probability must occur infinitely often—a principle that explains the 'infinite monkey theorem'.
Why this is interesting
Flip a fair coin repeatedly. It seems inevitable that heads will appear infinitely often, but can probability theory actually prove it? The Borel-Cantelli lemma gives a surprisingly simple answer—and it justifies the entire field of almost sure convergence.