Proof of infinite monkey theorem.

Question

I was just wondering, does the infinite monkey theorem also has a proof? And why is this called a theorem? It is sheer common sense. And what are its applications. I have heard about PHP and IEP and I also know that they are pretty useful. But what is the use of this and what is its proof?

Answer 1

The infinite monkey theorem states that if you have an infinite number of monkeys each hitting keys at random on typewriter keyboards then, with probability 1, one of them will type the complete works of William Shakespeare.

Let $A_n$ be the event that the $n^{th}$ monkey types the complete works of Shakespeare. Then if there are $m$ characters on the keyboard and $N$ characters in the complete works of Shakespeare, $\mathsf{P}(A_n) = m^{-N}$ for each $n$. Furthermore the $A_n$ are mutually independent. Hence, by the second Borel-Cantelli Lemma, since $$\sum_{n=1}^{\infty} \mathsf{P}(A_n) = \sum_{n=1}^{\infty} m^{-N} = \infty,$$ infinitely many of the events $A_n$ occur i.e. infinitely many monkeys will type the complete works of Shakespeare.