“The Paradox of Probability Theory: From Typing Monkeys to the Universe in Pi”

This is an astonishing paradox of probability theory. Imagine a monkey typing on a typewriter. What is the probability that he “randomly” produces a coherent word? And if he can randomly write a word, the animal should also be able to pull off the feat of typing an entire sentence, or even the few lines you are reading. It would be enough to wait long enough, but the thing must happen one day.

The first problem in this paradox is how much time you have to allow the monkey to write this article. If you give it a day, the probability will be very low, almost zero, if you give it a long time, even infinite, it will become one hundred percent. In other words, after such a long time, that it is even longer than the age of the universe, an infinite time, the macaque will have literally written the whole Divine Comedy of Dante. He will have written all the books already written and those yet to be written. The second problem is that the interesting pages will be drowned in a jumble of uninteresting pages. Even though you have them all on hand, you don’t have time to sort them out.

The history of the universe in the number Pi?

There are more modern versions of this paradox defined by Emile Borel (1871-1956). Imagine a sequence of numbers, for example your date of birth. What is the probability that said sequence is in the decimals of Pi? As a reminder, Pi, which used to be called Archimedes’ constant, is defined as the constant ratio between the circumference of a circle and its diameter. This number has an unlimited and non-periodic decimal expansion. The development of Pi starts with the famous 3.1415926… These three little suspension points hide a terrible, terrifying, incredible reality: there is no end to the decimals of Pi! We will therefore never be able to know them, since they are infinite. But we can calculate them…

The date of birth of the author of these lines, in the format DDMMYYYY, is found twice in the first two hundred million decimal places. On the other hand, the complete national register number is not there, and not in the first two billion either. Certainly, we will have to meet him afterwards. Certainly ? Are we so sure? To put it another way, if you pick a random number, will it be somewhere in the decimal places of Pi? The answer to this question fascinates mathematicians and it is not yet settled. We are leaning towards a positive answer.

In this case, Pi would be a universe number, because it would contain all possible numbers. If all the numbers were somewhere in the sequence of decimals of Pi, as long as we take the trouble to look far enough, we would then find there all the information of the galaxy, of the cluster of galaxies that encompasses us , all the information of the black holes, all the past and the future of our destiny and that of the neighbor.

All the “Libre Belgique”, yesterday and tomorrow

How to code everything in Pi? Imagine assigning a number to each of the letters of the alphabet without using the zero, a number for each of the punctuation marks, a number for making a line break and a page break… Imagine that the characters are separated by the number zero and the words by the double zero. With this method, you can digitally encode this entire chronicle, or even the entire diary you hold in your white hands. This sequence of digits is somewhere in Pi, the good old Pi. It therefore contains The Free Belgium and all the articles of the day. It contains Free yesterday and the day before yesterday. It contains all Libre since the creation of the newspaper in 1884 when it was still called The Patriot. So far, so good. But… But Pi therefore also contains the Libre tomorrow and the day after tomorrow, and each day following the other, it contains all the Libre to come and also all the newspapers of the competition which we will not quote. The future Goncourt is found in Pi, like the name of the next Prime Minister and the date of birth of your child who is not yet born…

What is the problem ? This information is unusable! If you want to read tomorrow’s paper in Pi, you might as well dip into the coffee grounds. In truth, even if you had easy access to all the decimals – which you don’t – you wouldn’t know where to start reading. You are pale blue chocolate. As of this writing, only the first 62.8 trillion decimal places, or trillions or even millions of millions, are calculated. The probability of finding the diary there is not great, it is even tiny.

The Library of Babel

Borges staged a narrow version of the paradox in his short story The Library of Babel. Imagine a library that contains all the books written by our friend the typing monkey. Borges limits books to 410 pages. Therefore, the number of works although very large becomes finite, since the number of possible combinations of a fixed number of characters on 410 pages is finite. The trick is that the librarian, over a hard life of hard work, never finds a single book of interest by the chance of his discoveries. Each time he dusts a collection, he comes across an unintelligible string of characters. Interesting books are drowned in the jumble of uninteresting books. The information is out there somewhere, but how do you find it? So the glory of the world.