Infinite Monkey Theorem

The Infinite Monkey Theorem is True

In 1913 Emile Borel proposed the thought experiment that a monkey hitting keys at random on a typewriter keyboard will – with absolute certainty – eventually type every book in France’s Bibliothèque nationale de France (National Library).
This is now popularly known as the infinite monkey theorem. Essentially, the monkeys become the Random Number Generator’s (RNG).
Let us test this. Imagine a line from a Shakespearean play, like this line from ‘As you like it’: “all the worlds a stage and all the men and women merely players they have their exits and their entrances and one man in his time plays many parts”. I have removed the upper case letters and punctuation (give the monkeys a break people!). As such it is 147 characters in length. Now imagine a typewriter with 27 keys (26 alphabet characters and one space, all the same size). Given enough time and monkeys, will the line ever get typed out? Of course it will! If we had enough time and monkeys they would eventually type out the entire works of Shakespeare!

But just how much time and how many monkeys do we have?

Some numbers

Let’s test our line from ‘As you like it’ being typed out by monkey’s randomly hitting away at our typewriter against our universe, which has been around for roughly 14 billion years. Imagine the monkeys bashing out 147 character line ‘trials’ on the 27 key typewriter at random. What is the chance that the monkeys type out the line, spaces included, perfectly (no upper case letter distinctions and no puntuation!)? The answer is 1/27¹⁴⁷. And after how many trials can we expect that it is more likely than not (ie. > 50% probability of having occurred) to have been typed out perfectly by our random monkeys? The answer*** is 27¹⁴⁷ log(2) ≈ 1.78e+210. So we need there to have occurred around 1.78e+210 opportunities for this event to be more likely to have actually happened than not.

Now that we have established how many trials we will need to have occurred before we feel confident that this line would have indeed been typed out randomly in a stepwise fashion, let us calculate whether we would have had enough chances for this event to occur in our universe given the age and number of possible chances that could have occurred. This is what is know as the available probabilistic resources.

Probabilistic Resources

Essentially we need to work out how many chances could have occurred since the dawn of time (the Big Bang) for our event to have occurred. If every particle in the observable universe (10⁸⁰) was subject to state changes at the rate of Plank time (10⁴³ trials per second, the maximum rate of change in nature) for the duration of the universe since the Big Bang (10¹⁷seconds) then there would only have occurred around 10^80+43+17 = 10¹⁴⁰ opportunities/trials so far. This falls way short of the required number of trials necessary for us to be confident that the above event is more likely to have occurred than not, since 1.78e+210 > 1.00e+140. The universe would have to be some 1.78e+69 times bigger than it currently is before we could say that the event is more likely to have occurred than not! If you would like to read more on the subject of small probabilities, then I highly recommend The Design Inference by Bill Dembski.

So in the end, to type out a single line from one work of Shakespeare with case and punctuation removed would take many more opportunities than the entire universe since the Big Bang has offered so far. What this means is that we don’t have enough probabilistic resources to expect to have seen such an event occur.

Well what about a functional protein?

Origin of Biological Information by Chance

In 2009 Dr Stephen Meyer released a death knell for Neo-Darwinism in his book ‘Signature in the Cell – DNA and the evidence for Intelligent Design’. The following experiment draws from this great book, more specifically from Chapter 9 and page 205.

The chance hypothesis for the origin of biological information is your only alternative if you are a materialist. Let us content ourselves for a moment with calculating the probability of producing just the information needed to build the suite of necessary proteins, bearing in mind that many more components would also be required and that therefore the calculation grossly underestimates the actual probability of building a living cell.

The simplest extant cell requires 482 proteins to perform its necessary functions. In minimal-complexity experiments scientists try to locate unnecessary genes or proteins in such simple life forms. Based upon these experiments scientists estimate that a simple one-celled organism might have been able to survive with as few as 250 – 400 genes. This has never been demonstrated though. Let’s take this speculation one step further in an effort to err massively on the side of caution and knock off a further 100 genes, leaving us with a very speculative 150 genes. Given 20 amino acids that are useful in building the protein chains necessary, we are staring at a 1 in 1/20¹⁵⁰ chance of finding the correct sequence of amino acids necessary (1.43e+195 ≈ 10¹⁹⁵). In other words, we need a universe 55 orders of magnitude bigger than our current one (10^140+55=195) in order to facilitate enough probabilistic resources to render the existence of this most simple living cell more likely than not. That is a staggering number, but if you thought that was big, how many resources do you think we would need for humans, who possess north of 20 000 genes, to come about by chance?

Conclusion

Well now, 10¹⁹⁵ is much greater than the probabilistic resources on offer (10¹⁴⁰⁾ by the entire universe since the beginning of time at the Big Bang. And this is on a very generous and conservative approach regarding the simplest extant cell today!If there wasn’t enough probabilistic resources to give us any confidence at all that even the simplest cell came about by chance, then we must simply reject the chance hypothesis. Why don’t you give the calculator a whirl yourself, try find scenarios where we do have enough probabilistic resources to perhaps observe the event. The calculator will give you an idea of the probabilistic resources at a few levels: the earth, the solar system, the Milky Way galaxy and the universe. Have fun!

*The values in the above table are reasonable estimates based on scientific evidence and represent the contempory accepted estimates for those items.
For example, the estimated number of particles in the universe is estimated to be between 10⁷⁸ and 10⁸².
**The estimated total number of events possible is a very generous number. Because it assumes that all the particles experience state changes to the tune of 10⁴⁵ changes per second and have done so for the entire existence of the item at hand, like the earth and so on.

***The Calculator

The calculator was designed to help develop your intuition regarding whether inference to chance or inference to design is a better adoption in a small probability setting. Specifically, the calculator calculates how many events we need in order for your event to have occurred such that it was more likely to have occured than not (ie the probability is >= 50% and not < 50%), which is then compared to local and universal upper bounds.

Prob (First event happens at or before attempt n) = 1 – (1-p)ⁿ

As long as p is not zero, this number can be made as small as you like by increasing the number n of attempts. This is essentially a special case of the law of large numbers, which (very) roughly says that things tend to average out in the end. We can easily work out how long we expect to wait to have, say, a 99% chance of generating Shakespeare. Suppose we want a chance F of being finished by attempt n. Then

F = 1 – (1-p)ⁿ and ∴ n = log(1-F)/log(1-p)

where the logarithms can be to any base, but we shall assume natural logarithms to base e. For small p we can get a convenient approximation, since

log(1-p) ≈ -p and ∴ n ≈ -log(1-F)/p

We can express this in an even simpler way. Suppose p = 1/T, that is there is a 1 in T chance of the event occurring. And suppose we want to be really sure of observing the event, so that

F = 1 – 1/K

where K is large. That is, we want there to be only a 1 in K chance of still being waiting after time n. Then

n ≈ T(log K)

So if you have an event with a 1 in 1024 chance of occurring (like a trial of 10 heads in a row = 1/2,¹⁰), and you want to have only a 1 in 2 chance of still waiting by time n, then set n to be 1,024 log(2) ≈ 700. This means that there is a 50% chance that the event will have occurred before ≈ 700 attempts (trials of 10 flips).