05/06/2019

In Folha, the difficulty in understanding probability.

Reproduction of Marcelo Viana's column in Folha de S.Paulo.

Homo sapiens is a remarkable machine. Honed 200,000 years ago on the African savanna, our hardware and software have become surprisingly flexible.

Riding a bicycle or writing, for example, are things we do well, even though they were useless for the survival of our ancestors. But there is one area where we are bad: understanding probabilities. I will show the reader.

Two babies were born in the neighborhood, and we know that one of them is a girl. What is the probability that both are girls? Most people answer that it is ½ (50%), arguing that the other baby could be a boy or a girl, and that these two cases are equally likely. The answer is wrong!

Initially there are 4 possibilities: (boy, boy), (girl, boy), (boy, girl) and (girl, girl). Since we know that one of the babies is a girl, the first case is excluded. That leaves three, all equally likely, of which only one corresponds to two girls. Therefore, the correct probability is 1/3.

One of the most popular examples is the Birthday Paradox. In a class of 25 students, what is the probability that two of them will have their birthday on the same day? Most people believe it to be small, after all there are 365 days in a year and few students. But the correct answer is 56%.

Another example was popularized by game shows. On stage there are three doors: behind one is a prize and behind the others, something bad. The player chooses one, but doesn't open it. The host opens another door—necessarily a bad one—and asks the player if they stand by their initial choice. The answer is anything but intuitive. Even without knowing which door is the good one, the contestant should always switch: it can be proven that the probability of winning the prize doubles.

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