If a monkey was placed before a typewriter and incentivized to hit keys at random it would take it vastly longer than the universe’s anticipated lifetime to produce the complete works of Shakespeare, or even a children’s book.
The Infinite Monkey Theorem is a thought experiment used to epitomize the way random processes can produce something very unlikely if left long enough. It’s been framed in several ways. Sometimes one monkey is given a typewriter and hits the keys randomly for an infinite amount of time. Sometimes the number of monkeys is also infinite. Usually, the goal is all Shakespeare’s plays and sonnets, but sometimes the proposer will settle for Hamlet.
The idea has worked its way so deeply into popular culture that it has inspired countless references, such as in The Hitchhiker’s Guide to the Galaxy, and even the name of one of the world’s most popular science podcasts. Now two mathematicians have considered the prospects when time is made finite.
Dr Stephen Woodcock and Jay Falletta of the University of Technology, Sydney, calculated the probability of a monkey hitting the letters required in their appropriate order. They concluded that even if the monkey didn’t take lunch breaks or sleep, it would be lucky to get even the first line of a play.
“The Infinite Monkey Theorem only considers the infinite limit, with either an infinite number of monkeys or an infinite time period of monkey labour,” Woodcock said in a statement. “We decided to look at the probability of a given string of letters being typed by a finite number of monkeys within a finite time period consistent with estimates for the lifespan of our universe.”
Woodcock and Falleta’s “Finite Monkey Theorem” requires some assumptions. The authors decided to make things easier for the monkey by removing such extraneous keys as those for numbers and punctuation marks unknown to Shakespeare, and didn’t worry about capitalization. Keys on their 30-character keyboard were assumed to be struck randomly, rather than some being favored. The pair also calculated the chances of more modest goals being achieved.
The monkeys were assumed to type at a rate of one keystroke a second – slow by the standards of a good human typist but possibly requiring drug enhancement for a creature with smaller hands. The calculations are more complex than might be assumed, as the authors address the risks of factors such as overcounting, where a desired sequence is achieved more than once.
Computer-simulated monkeys, where only human intervention got the computerized monkeys far, demonstrate any progress will be painfully slow, while live monkeys perform even worse.
The mathematics makes clear that a single monkey would almost certainly not get even the first act of Hamlet written before the anticipated heat death of the universe, 10100 years away. Not all physicists agree that the perfect entropy known as heat death will be the universe’s eventual fate, but since most alternative scenarios would come even earlier, these would only make things worse.
Consequently, Woodcock and Falletta chose to call in reinforcements. Unfortunately, much as we love the rest of their work, at this point they choose to abandon science. Having apparently failed to consult a primatologist, the pair call chimpanzees “monkeys” – which even Planet of the Apes knew is offensive – and enslave the entire global population of chimps into typewriter duty. We hope the Jane Goodall Institute will send a letter of protest both for mental cruelty and scientific inaccuracy.
Based on 200,000 living chimpanzees and a lifespan of 30 years, and assuming the chimp population somehow remains stable without taking time off typing for sex, this gives the chimpanzees 6.4 x 10103 lifespans to work with, each lasting 109 seconds.
On this basis about five percent of the enlisted chimps would produce the word “bananas” at some stage in their typing, at which point we hope they would be rewarded with several and time to enjoy them.
On the other hand, the chance of producing Hamlet, let alone complete works of Shakespeare, is effectively zero.
Even the 1,800 words of Curious George would be so unlikely to come off the typewriters of the chimpanzee population before the universe dies as to be effectively impossible: less than one chance in 1015,042. “We can see that all but the most trivial of phrases will, in fact, almost certainly never be produced during the lifespan of our universe,” the authors write.
This might be considered another reason to restore chimpanzee numbers, but the authors conclude, “It is not plausible that, even with possible improved typing speeds or an increase in chimpanzee populations, these orders of magnitude can be spanned to the point that monkey labor will ever be a viable tool for developing written works of anything beyond the trivial.”
In a tradition stretching all the way back to Zeno’s paradox, the authors find that an outcome certain in the infinite (whether of monkeys or times) is almost impossible in the finite case.
AI may be a threat to human creativity, the authors conclude, but monkeys are not, unless they learn something from their time at the typewriter, and their output ceases to be random.
The study is published in Franklin Open, a journal reviewed by human, not monkey, peers.
