The probability that a solitary, immortal monkey, typing randomly on a standard, unbreakable keyboard, would eventually produce one of Shakespeare's plays is estimated. The question of what age of the universe, if any, would be large enough to give the monkey a 50% chance of success, is investigated.

The monkey problem is believed to have originated with a statement attributed to Huxley:

Six monkeys, set to strum unintelligently on typewriters for millions of years, would be bound in time to write all the books in the British Museum..

and is quoted by J. Jeans in Mysterious Universe, Cambridge University Press, 1930, p.4. As Jeans was an accomplished physicist, it is surprising he does not either prove or disprove the claim. Perhaps because Huxley did not specify exactly how many millions of years he was thinking of, Jeans let the remark go unchallenged.

The exact age of the universe one finds in the literature can be expected to vary as more accurate astronomical data is obtained by scientists. The value we will use is that currently given by NASA. The number of letters in a Shakespearean play is also an approximation. However, only order of magnitude accuracy is required to perform the calculation, so the approximations are reasonable.

1. \( {\text{Age of Universe = 13.7 billion years =}} 10^{17} s\)   ^[1]
2. \( {\text{ Number of chars/letters in a Shakespeare Play = }} 10^5 \) (Rough approximation)
3. The keyboard contains 44 keys (Again, this varies, depending on the keyboard)
4. All keys are equally likely to be struck by the monkey. (Since the space bar is a much larger key than the others, in reality the monkey would be expeced to strike it more often. We ignore that nuance.)
5. Ignore upper and lower case distinctions
6. The monkey types at a rate of one char per second

In general, the calculation of a probability reduces to solving two counting problems. First, you have to calculate the number of elements in the sample space. Second, you have to calculate the number of elements in the event. A fundamental technique that often enables us to do this is the Product Rule

Let our budding playwright monkey type precisely \( 10^5 \) characters. Let's enquire as to the probability that the monkey in fact produced a Shakespearean play. To do so, we will need to count the elements in the sample space, and the elements in the event.

If an event or experiment can only have two outcomes (success and failure), then independent repeated trials of that event are called Bernoulli trials. A standard example of a Bernoulli trial is the tossing of a coin. Here is the essential approximation in our heuristic approach to the Monkey-Shakespeare problem -- every \(10^5 \) characters typed by the monkey will be considered to be a Bernoulli trial. That is, we let the monkey type \(10^5 \) characters (the assumed number of characters in the Shakespeare play) and then ask if the given Shakespearean play was, in fact, produced (success) or not (failure). By approximating the problem as a sequence of Bernoulli trials, we can apply well known and standard results of elementary probability theory to get a heuristic answer to our basic question.

First, we review the basic facts of Bernoulli trials. Denote the probability of success of a Bernoulli trial by p, and the probability of failure as q, with p + q = 1. Then, we have the following result for the probability of k successes in n trials:

P (k successes) = \(n\choose k\) \(p^k q^{n-k} \)

and,

P (1 or more success) = 1 - \( q^n \)
where \(n\choose k\) is defined as \( \frac{n!}{(n-k)!k!}\)

Setting \( np = \frac{1}{2} \), we get \( n = \frac{1}{2p} \), so that the number of trials n to give the monkey a 50% chance of success is
\(n_{50} = .5 * 10^{164344} = 5 * 10^{164343}\)
and so,

Thus we see that the required age of the universe for a reasonable chance of monkey success is a surprisingly huge number relative to the known age \(10^{17} s \) of the universe.

In fairness to Huxley and Jeans, I must point out that before Einstein's General Theory of Relativity, and the resulting Big Bang theory, time was considered to be absolute, and infinite. Therefore, one did not typically consider the age of the universe in one's calculations, and tremendously large stretches of time were not necessarily unrealistic.

In conclusion, I must point out why this is merely a heuristic calculation. It has been assumed that we can model the calculation by a series of Bernoulli trials. This approximation has the effect of under-estimating the probability. It would be possible, for example, for the monkey to have successfully reproduced the play after \( 10^5 + 1 \) chars (i.e.,1 trial plus the first char from the next trial) have been typed... However, approximating by a series of Bernoulli trials does have the advantage of using well known and elementary counting techniques to make obvious the rather absurd nature (absurd given our current knowledge of the finite age of the universe) of Huxley's original statement.