Topics Mathematics

Question If an infinite number of monkeys were at an infinite number of typewriters, would the work of Shakespeare eventually come out?

Accepted:
October 27, 2005

Comments

The answer is that the work of Shakespeare would almost surely come out. More precisely, the probability that at least one of them will type Shakespeare is 1. This doesn't mean that it is absolutely certain to happen. It means that if you did the experiment repeatedly, and kept track of how many times at least one monkey typed Shakespeare and how many times none of them did, you would expect that the fraction of the time that at least one monkey typed Shakespeare would approach 1. It could occasionally happen by chance that no monkey typed Shakespeare, but if you kept repeating the experiment you would expect this to happen so infrequently that in the long run, the fraction of the time that no monkey typed Shakespeare would approach 0.

To simplify things, let's assume that a monkey types 18 random characters, and each character is either one of the 26 letters or a space. One 18-character string that the monkey could type is "to be or not to be", but of course there are many others. The number of possible strings is 2718, which is about 5.8 x 1025. Let's call this number c. Then the probability that a single monkey will type "to be or not to be" is 1/c, which is very close to 0, and the probability that he will type something else is 1 - 1/c, which is very close to 1. But now suppose you have c monkeys, each typing 18 random characters. The probability that none of them types "to be or not to be" is (1-1/c)c. It is a nice calculus exercise to verify that for any large number c, (1-1/c)c is approximately equal to 1/e, which is about 0.368. So the probability that no monkey types "to be or not to be" is about 0.368, and therefore the probability that at least one monkey types "to be or not to be" is about 0.632. As the number of monkeys increases, the probability that no monkey types "to be or not to be" keeps decreasing. With 2c monkeys, the probability that no monkey types "to be or not to be" is (0.368)2, which is about 0.135, so the probability that at least one monkey types "to be or not to be" is about 0.865. With 10c monkeys (about 5.8 x 1026), the probability of getting "to be or not to be" is more than 0.9999. With infinitely many monkeys, the probability is 1.

They couldn't produce the work of Shakespeare. Only Shakespearecould. In fact, there's good reason to think that they couldn't produceany work of literature at all. Doing that would require sorts ofintentions that those infinite monkeys would not have.

ProfessorVelleman's reasoning shows that there's a probability of 1 that theywould produce a text identical to the text of one of Shakespeare'sworks. Actually, it looks like there's a probability of 1 that theywould produce texts identical to that of all of Shakespeare's works (and allversions of those works). But texts are not identical to literaryworks--Shakespearean or otherwise. A word-for-word duplicate ofDicken's Bleak House written by, um, Shmickens would not be Bleak House.See Jorge Luis Borges' wonderful story "Pierre Menard, Author of theQuixote" for some relevant thoughts on the text/work distinction.

Question If an infinite number of monkeys were at an infinite number of typewriters, would the work of Shakespeare eventually come out? Accepted:October 27, 2005

Question If an infinite number of monkeys were at an infinite number of typewriters, would the work of Shakespeare eventually come out?

Accepted:
October 27, 2005