30/10/2019

In Folha, Viana addresses paradoxes of probability.

Foto: Anders Adermark/Flickr

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

The word "dice" in Latin is "alea," which gave rise to "aleatory." But rolling a die isn't truly random: if we knew exactly how it's done, we could, in principle, calculate the die's movement and predict which face will come up. Since this isn't feasible in practice, it's more useful to think that any of the six faces can come up, at random, with equal probability.

Thus, probability has a lot to do with ignorance: if we were omniscient, every event would have a probability of 0 (impossible) or 1 (inevitable). Therefore, probabilities can change based on additional information. A simple example: initially, the probability of rolling a 5 is 1/6, but knowing that an odd number was rolled, this probability becomes 1/3.

However, sometimes new information is subtle, leading to counter-intuitive conclusions. The following two examples, which I learned from Professor Nicolau Saldanha of PUC-Rio, are particularly intriguing. Solutions are welcome via email at viana.folhasp@gmail.com .

A piece of furniture has three drawers. One drawer contains two white t-shirts, another two black t-shirts, and the third one white and one black t-shirt. We open a drawer at random and take out one of the two t-shirts at random, without looking at the other. The t-shirt we take out is white. What is the probability that the other t-shirt, which was left alone in the drawer, is also white?

Answer A: The three drawers are equally likely, but we know we didn't choose the one containing only black t-shirts. Therefore, the probability of having chosen the drawer with two white t-shirts is 1/2. Answer B: The six t-shirts are equally likely, but we know we didn't choose black. That leaves the three white ones, all with a probability of 1/3. In two cases, the companion t-shirt is also white, so the probability is 2/3. Which is correct, A or B?

Andrei won a prize in a raffle, but he needs to choose between two identical sealed envelopes. Each envelope contains a check: Andrei only knows that the value of one is double the value of the other. He opens one envelope and sees that the check is for R$100. What is better to maximize his winnings: keep this one, or exchange it for the other?

To read the full text, visit the newspaper's website or check the print version .