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The probability is not a monster

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The probability is not a monster

By Gabriela Cybis

Our intuition is notoriously flawed in assessments that involve uncertainty. Why?

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The auditorium show host shows you three closed doors. Behind one of them is the sports car of your dreams. Behind each of the other two, a monster. You choose one. "Are you sure?" Asks the presenter. He will give a hint. Open one of the two doors you didn't choose and a monster appears. "Now there are only two doors left," he says. "Do you keep your choice or do you want to switch?" And now, what to do? Does it make a difference in terms of probabilities?

This charade, known as the Monty Hall problem in honor of the homonymous American presenter, was published in an American Sunday newspaper in 1990, causing a stir. More than 10,000 readers wrote to the magazine questioning the solution presented, an impressive number considering the (dis) interest of the general public for mathematical problems. Even the famous mathematician Paul Erdös was not convinced of the answer, even given the clear demonstration that proves it. The riddle illustrates how our intuition can lead us to false beliefs, especially on issues related to probabilities.

Human beings are notoriously flawed in assessments that involve uncertainty. For example: who are more dangerous, mosquitoes or sharks? Mosquito bites are responsible for transmitting diseases to millions of people, of whom around 700,000 die worldwide each year. In contrast, in the same time, sharks are responsible for around ten deaths. Therefore, mosquitoes are more dangerous than sharks, but we continue to fear these predators much more.

This makes sense from an evolutionary point of view: when there is a risk of sharks in the water, it is healthy to be afraid. While this strong intuitive response has served us well in a wilder past, today the conditions of life are more complex. In a world where we get sick slowly due to exposure to carcinogens or excessive consumption of fat, among others, these simple instincts do not always serve as a reliable guide. We often lack more refined notions of risk.

We call cognitive bias those situations in which people systematically make errors of judgment. There are a number of them linked to probability, with names like “player fallacy” and “bias of negligence of probability”. They have been known for decades by the scientific community, which conducts experiments with groups of volunteers who are exposed to pranks built to evidence such bias.

This knowledge can be used in the most effective communication of data involving uncertainties, improving the quality of decisions by health professionals or in the business world. It can also be exploited to induce people to make decisions in the interests of others. This does not mean, however, that we are bound to make the same mistakes again and again. Knowing the most common mistakes and being familiar with logical and mathematical reasoning are strong allies to avoid them, although they do not completely eliminate them.

And the solution to the Monty Hall problem, do you already know what it is? Most people say that it makes no difference whether or not to change the door after the presenter's intervention. After all, all doors are just as likely to bring the car, right? Wrong. The act of revealing the content of one of the doors after its initial choice alters the probabilities of the second stage of the problem. Thus, changing the door after the presenter's intervention is the best strategy to conquer the car of your dreams. Why?

To verify this, we can divide the problem into two cases.

Case 1: Let's say that in the first step you choose the right door (this occurs with probability 1/3); in this case, when you change doors in the second step, you will always end up with the wrong door.

Case 2: Let's say that initially you choose one of the wrong doors (what happens with probability 2/3); in this case, when the presenter removes a door with a monster, only the correct door will remain.

Therefore, we conclude that the strategy of changing the door in the second stage gives you the best chance of returning home with your big car, probability 2/3, to be precise.

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Gabriela Cybis is a biologist, professor of statistics at UFRGS, working in statistical modeling for genetics.

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