Discovering the concept of infinity and probability

“How can a rock stem the sea? ” used to sing the legendary Italian singer Lucio Battisti.

“How can a monkey write the Divine Comedy?”, I ask you today … you’ll easily answer: “It cannot!” And here comes the fun: Emile Borel did not think so!

Today I’ll show you that not only a monkey can write the Divine Comedy, but rather that, in certain circumstances (actually only one), it will almost certainly write the Divine Comedy (in mathematical terms it means that the event has got probability equal to 1). Curious to know what is the powerful condition that makes a monkey write our beloved text with near full certainty. The only condition is that time available is infinite.

Our monkey has got at its disposal a keyboard of m characters all with the same probability of being pressed equal to 1/m. The probability of intersection of independent events is equal to the product of the individual probabilities, so the probability that our monkey writes any text, combination of k elements, is equal to (1/m)^k that, however small, is strictly greater than 0.

Still with me? Let’s carry on! Given an event with probability x, the probability of its complementary event is 1-x. The probability not to write our text, composed of k elements, is given by 1-(1/m^k). Now let’s imagine that our monkey can do n attempts to write the text: the probability of never writing the text is reduced to (1-1/m^k)^n. If we now make n tend to infinity, this probability goes to 0. So the probability that our monkey writes our text of k characters having at disposal an infinite number of attempts is exactly equal to 1. As we wanted to demonstrate our monkey, with infinite time at disposal, will almost certainly write the Divine Comedy!

What we have shown is the so-called theorem of The infinite monkey. A curious note: the text of the theorem varies from country to country. The British, for example use the Hamlet while the French, in their grandeur, consider all the volumes of the French National Library!

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