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Sunday, July 3, 2022

D’Alembert and the theory of probability

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Jean le Rond d’Alembert.

With his absurd law of reproduction, the macho king only managed to drastically reduce the number of his subjects last week, but the proportion of men and women remained the same. Consider a sample of 100 couples beginning to conceive: Among their firstborn are approximately 50 boys and 50 girls; the parents of the latter will no longer be able to have more offspring; but the other 50 couples, assuming they all continue to breed, will have about 25 secondborn boys and 25 girls, and so on, indefinitely? No: In principle, only 12 or 13 couples will have a third child, only 6 or 7 will have a fourth, only 3 or 4 will have a fifth, only 1 or 2 will have a sixth and perhaps none will have a seventh. Each “offspring” will be half the size of the previous one (and soon the birth rate will be zero), but in all of them there will be about as many boys as girls.

Despite its simplicity, the problem sparked a wide debate, within the framework of which Juan José Rodríguez d’Alemberts advanced criticism of conventional probability theory, which, in his opinion, did not pay due attention to experience in formulating its assumptions and definitions. Which, by the way, makes it harder to understand that the great French mathematician and encyclopedist incorrectly estimated that the probability of getting at least one head when tossing two coins in the air is 2/3. What is the true probability and what misleading reasoning leads to the value 2/3? Interestingly, this misjudgment is a clear example of the disregard for real situations that d’Alembert himself criticized.

An example of how complicated it can sometimes be to formulate a probability problem adequately can be found in Bertrand’s paradox (I am not referring to his famous economic paradox related to the Nash equilibrium, but to the probabilistic paradox that we discussed a few years ago treated time) . some years):

What is the probability that a chord drawn at random in a circle is larger than the side of the equilateral triangle inscribed in it? Depending on how the problem is approached, we can arrive at different answers: 1/2, 1/3, 1/4… and I invite my attentive readers to reconstruct the reasoning that leads us to these or others values. For example, which of the above estimates does the attached figure suggest and why?

The Haggadah of the Sweaty Wanderers

Oral tradition offers us numerous and instructive examples of flawed, inadequate, misleading, or implausible approaches. In this sense, a well-known Haggadah (fairy tale or fable with a moral from the Hebrew tradition) speaks of two wayfarers who, a rabbi tells his disciple, are walking along a dusty path on a hot summer’s day. Finally they find a source to quench their thirst. One of the walkers’ face is dirty with sweat and dust, while the other’s face is dry and clean. Which of the two is washing their face? the rabbi asks.

The naïve answer is that whoever has a dirty face washes his face. The brilliant answer is that whoever has it clean washes it because everyone sees their partner’s face and not their own, so whoever is clean believes they are dirty too and vice versa. But what would be the realistic answer in d’Alembert’s (and the Rabbi of the Haggadah) critical line?

Carlo Frabetti is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 popular scientific works for adults, children and young people, including “Maldita Physics”, “Malditas Matematicas” and “El Gran Juego”. He was the screenwriter for The Crystal Ball.

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Source elpais.com

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