If there were as many monkeys as there are atoms in the observable universe typing extremely fast for trillions of times the life of the universe, the probability of the monkeys replicating a single page of Shakespeare is unfathomably small.
Ignoring punctuation, spacing, and capitalisation, a monkey typing letters uniformly at random has a chance of 1 in 26 of correctly typing the first letter of Hamlet. It has a chance of 1 in 676 (26 x 26) of typing the first two letters. Because the probability shrinks exponentially, at 20 letters it already has only a chance of 1 in 26^20 = 19,928,148,895,209,409,152,340,197,376. In the case of the entire text of Hamlet, the probabilities astonishingly small, and inconceivable. The text of Hamlet contains ~130,000 letters. Thus there is a probability of 1 in 3.4 x 10^183,946 to get the text right at the first trial. The average number of letters that needs to be typed until the text appears is also 3.4 x 10183,946, or including punctuation, 4.4 x 10^360,783.
If every proton in the observable universe were a monkey with a typewriter, typing from the Big Bang until the end of the universe (when protons might no longer exist), they would need a still far greater amount of time - more than three hundred and sixty thousand orders of magnitude longer - to have a 1 in 10^500 chance of success.
For a 1 in a trillion chance of success, there would need to be 10^360,641 universes made of atomic monkeys. As Kittel and Kroemer put it in their textbook on thermodynamics, the field whose statistical foundations motivated the first known expositions of typing monkeys, "The probability of Hamlet is therefore zero in any operational sense of an event...", and stating that the monkeys must eventually succeed "gives a misleading conclusion about very, very large numbers."
There is less than a 1 in a trillion chance of success that such a universe made of monkeys could type any particular document a mere 79 characters long.
