The infinite monkey theorem

Infinite monkey theorem

It is often said, that if a million monkeys type on their keyboards, they will eventually type out the entire works of Shakespeare. This is an interesting theorem and takes us into the interesting realm of ridiculously huge numbers. But before we delve into this, let us first look at an example closer to home – an ordinary bicycle lock, where there are three rings, each which has the numbers from zero to nine. If your friend tells you he forgot the combination to his lock, but managed to open it by trying a few combinations by random, you’d probably believe him but consider him very lucky. In this case the number of possible combinations is 10³ or 10*10*10 or 1000. Thus the probability of opening the lock by trying at random is 1/1000, that is very possible indeed. But let’s imagine your friend told you he had opened a lock with five rings. Now you might wonder a bit, because in this case the number of combinations is 10⁵ or 100000. If your friend had not lied before, you might believe him, but you probably would raise an eyebrow. Let’s take this futher and imagine your friend told you he managed to open a lock with ten rings by trying a random combination. Now the probability of opening the lock is 1/10¹⁰ or 1/10000000000 – you’d more likely win the lottery. But every week someone wins the lottery, so in theory your friend may have managed to open the lock. Let’s take this even further and imagine our friend tells us he managed to open a lock with a hundred rings. Now the number of combinations is 10¹⁰⁰ – and incredibly ridiculously huge number, a one followed by a hundred zeros. To get an idea how huge this number is, consider there are only 10⁸² atoms in the known universe. The probability of opening a lock with 100 rings is 1/10¹⁰⁰ – you are more likely to find one marked atom in the whole universe. In this case you would be certain your friend had lied. Only a fool would believe your friend. Sure that event is possibly in theory, but only in theory, in practice no one on this earth would be able to open that lock by a random search, not even if they tried from the Big Bang till the end of the universe.

Let’s get back to the monkeys. Ignoring punctuation, the probability of the monkeys typing the first 100 characters of Hamlet would be 1/26¹⁰⁰ or 1/(3*10¹⁴¹). As in the example of the bicycle lock with 100 rings, so in this case also you’d more likely find a single atom in the universe than randomly type the first 100 characters of Hamlet. If one hundred trillion monkeys had been typing 100 trillion characters per second since the big bang, they would have typed 10¹⁴*4*10¹⁷*10¹⁴=4*10⁴⁵ characters, which according to my calculator is exactly zero percentage of all the possible combinations (3*10¹⁴¹). So with certainty the monkeys could not have typed even the first on hundred characters of Shakespeare, let alone the entire works.

Model of the newly-identified clorophyll f enzyme that converts chlorophyll a into chlorophyll f. The green molecules near the bottom of the structure represent chlorophyll a molecules that would be modified in light to produce chlorophyll f.

The infinite monkey theorem is often cited as proof that evolution is possible, because if evolution has millions of years to work with, even unlikely events are possible. So let’s look at the oring of life, what is the probability of the first living cell arising from the primordial soup? We shall not consider here an entire cell, but we’ll only look at a single protein. Life is mostly made up of proteins. Proteins are polypeptide chains, that are made up of twenty different aminoacids. On average a protein is made up of 300 aminoacids. Therefore there are a 20³⁰⁰, that is 2*10³⁹⁰, possible ways to build up a 300 aminoacids long protein. The probability of finding one such protein by random is thus 1/(2*10³⁹⁰), an incredibly ludicrously small number.

But, you may object, that is just the probability of finding one protein, we need to consider the number of all functional proteins. You are correct. How many are there? Douglas Axe in the year 2004 estimated that there are one in 10⁷⁷ of functional proteins to junk. In the year 2012 Durston and Chiu estimated that number to be one in 10¹⁰⁰. Surely these are just estimates, it is impossible to imperically test all the possible functional proteins. What is clear though, is that functional proteins are very rare. If every single drop of water in our oceans (2.7*10²⁵ drops) since the Big Bang (4.4*10¹⁷ seconds) had tried to produce functional proteins once per second, they would have traversed through 10⁴³ different proteins – not even close to likely, that they would have found a single functional protein. Dispite this all the biology books claim that life was spontaneously created in the primordial soup.

Long tailed weasel

Richard Dawkins steps in. His brilliant Weasel-program solves this problem. His program shows, that in only 40 generations his program can produce the sentence “ME THINKS IT IS LIKE A WEASEL”. How is that possible? Because his program compares the randomly generated characters and only chooses the ones that match the given pattern. But this is information that evolution does not have. Evolution can not know if two aminoacids are in the right order, it can not know, if the first ten aminoacids are in the right place, it can’t know if 99% of the aminoacids were in the right sequence. Contrary to what Dawkins claims, there is not shortcut to the mount improbable. Nobody has ever demonstrated a path from one protein to another. There are no such paths. There is only an infinitesimally small chance of finding even a single protein. And if we are talking about the origin of life, then there is no mechanism of evolution that could have operated at that time, before the dawn of the first living creature.

But, every now and then, somebody wins in the lottery. But nobody wins a million times in a row.

Douglas Axe, ” Estimating the Prevalence of Protein Sequences Adopting Functional Enzyme Folds”, 2004, Journal of Molecular Biology

Kirk Durston and David K. Chiu, “Functional sequence complexity in biopolymers” (2012)

10⁸² atoms in the universe.
4.35454*10¹⁷ seconds since the Big Bang.
2.664*10²⁵ drops of water in the oceans.