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The so-called 'birthday paradox', what it is and how it is explained

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The word paradox comes from the Greek for 'against' and doxo 'opinion', so that etymologically we could say that it is that which is incredible or absurd. In mathematics and physics we find very famous paradoxes: that of the twins, that of the grandfather, that of Schrödinger's cat, that of Möbius or Monty Hall. Today we are going to deal with a paradox that is not really a paradox.

First of all, a trivia question. How many people do you think you need to gather in a group so that the probability that two people have the same birthday (day and month) is greater than 50 percent? 50, 60, 70… people?

Surely many of us will think a priori that we have to gather a very large number, however , with only 23 people a probability of 50.7% is reached.

A figure that is it rises to 99.6% when there are 57 people in the meeting.

Let's go to a specific case, for example, our monarchy. Of a total of nineteen Spanish monarchs -those included between the Catholic Monarchs and Felipe VI- there are two coincidences: Carlos II and Carlos IV, both were born on November 11, and José I and Juan Carlos I, who were born on January 5.

The dovecote principle

This probability is known as the 'birthday paradox', a term that, honestly, does not seem very appropriate, since in the strict sense of paradox it has nothing, rather it is a logical contradiction to intuition and that is in line with the so-called 'loft principle'.

This principle states that if 'n' pigeons are distributed in 'm' lofts, and if 'n' is greater than 'm' -there are more pigeons than lofts-, there will be at least one loft with more than one pigeon. So far everything is in order, it seems such a trivial and simple matter that it does not even need a demonstration, however, it has great power and numerous applications that can be used in graph theory, in combinatorics and in computer science.

The first to approach this concept was the German Johann Peter Gustav Lejeune Dirichlet (1805-1859), to whom the modern formal definition of a function. This Teutonic mathematician baptized it as the 'box principle': Schubfanchprinzip (1834).

Some curious applications

Extrapolating the birthday paradox, we can affirm that in a concert with more than 800 people there will be, at least, two of them who will have the same initials of their name and your first surname. This is because each initial is one of the twenty-seven letters of the alphabet, which means that there are 27×27=729 possible combinations between names and surnames.

It has been several decades since Martin Gardner posed his famous sock math problems. One of them said that if there are 10 black socks and 10 white socks in a bag, how many would you have to take out to find a pair that matches in color? Simply, three.

Another curious way of posing the problem is to analyze whether the last two numbers of the license plate are repeated in fifteen randomly noted cars . The probability is 67%, that is, we will win two out of three times. A figure that can be raised to five out of six times if instead of fifteen license plates nineteen are analyzed.

A problem with a little more substance Mathematics would be to show that if five integers are taken between 1 and 8 -both inclusive- there will necessarily be an even number of them that add up to 9. First of all, there are four integer pairs between 1 and 8 that add up to 9, namely, 1- 8, 2-7, 3-6 and 4-5. These pairs would be the 'dovecotes' and the five numbers taken are the 'doves', so according to the dovecote principle there must be at least two numbers that belong to the same pair.

All these examples are very elementary problems, but Dirichlet's principle can be applied to much more complex approaches.

Pedro Gargantilla is an internist at El Escorial Hospital (Madrid) and the author of several popular books.

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