Statistics and Monkeys on Typewriters

Here are the first two sentences of the prologue to Shakespeare’s Romeo and Juliet:

Two households, both alike in dignity,
In fair Verona, where we lay our scene

This excerpt has 77 characters. Now we let a monkey start typing random letters on a typewriter. Once he typed 77 characters, we change the sheet and let him start over. How many tries does he need to randomly reproduce the above paragraph?

There are 26 letters in the English alphabet and since he’ll be needing the comma and space, we’ll include those as well. So there’s a 1/28 chance of getting the first character right. Same goes for the second character, third character, etc … Because he’s typing randomly, the chance of getting a character right is independent of what preceded it. So we can just start multiplying:

p(reproduce) = 1/28 · 1/28 · … · 1/28 = (1/28)^77

The result is about 4 times ten to the power of -112. This is a ridiculously small chance! Even if he was able to complete one quadrillion tries per millisecond, it would most likely take him considerably longer than the estimated age of the universe to reproduce these two sentences.

Now what about the first word? It has only three letters, so he should be able to get at least this part in a short time. The chance of randomly reproducing the word “two” is:

p(reproduce) = 1/26 · 1/26 · 1/26 = (1/26)^3

Note that I dropped the comma and space as a choice, so now there’s a 1 in 26 chance to get a character right. The result is 5.7 times ten to the power of -5, which is about a 1 in 17500 chance. Even a slower monkey could easily get that done within a year, but I guess it’s still best to stick to human writers.

.This was an excerpt from the ebook “Statistical Snacks. Liked the excerpt? Get the book here: Want more excerpts? Check out The Probability of Becoming a Homicide Victim and Missile Accuracy (CEP).


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