Given the Innite Monkey problem forget about the word banana

Given the Innite Monkey problem, forget about the word banana. Take the text of Romeo and Juliet which has 149,248 characters. Also assume that it takes 1/2 a year for the monkey to type that many characters. What is the probability that the monkey will type Romeo and Juliet in 100 billion years?

Solution

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.

Suppose that the typewriter has 50 keys and the monkey as has to type 149,248 characters. Therefore, the probability of the monkey typing the first 149,248 desired character is (1/50)^149,248, since each keys are pressed independently.

Since the monkey can type 149,248 characters in 1/2 year, it can type (100 billion x 2 x 149,248) characters in 100 billion years.

Therefore, the probability of the monkey typing the first 149,248 desired character is (1/50)^{149,248 x 2 x 100 billion} which is very small, but not zero.