Monty Hall Problem

Dmitry Khlestkin


Monty Hall Problem

Monty Hall Problem is the best example of people’s inability to correctly weigh up the chances for success, to choose a favourable outcome and put off the two unfavourable.

This is crucial for punters, too. After all, if a person is not able to determine the probability of an outcome looking at the odds and doesn’t understand the value of the bet, he won’t be able to keep favourable balance in the long run.

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A new fancy car is behind one of three doors. Behind the other two there are goats. You have to guess which door the car is behind. However, you do not have the knowledge required to help you understand what is behind the doors. You choose a door, you watch it open, and see a goat behind it. Now you need to decide whether to stick to your initial pick or choose a different door? Monty Hall problem was named after the anchor of a popular TV show in the United States of 60-70s, Let’s Make a Deal. It is a simple mathematical puzzle effectively demonstrates how people find it difficult to make, one would think, a simple choice.

With this simple but tricky puzzle, the US TV show demonstrated how an average person follows his intuition when faced a choice based on probabilities. The same behavior could be observed among common punters. When the puzzle was published in Parade magazine, about 10,000 readers complained that the answer assumed correct by a magazine, was wrong. Among the non-believers were several professors of mathematics.

Solving the problem

Monty Hall problem may be simply solved in your favour – always switch the picked door! After opening the first door, with the goat behind it, it is clear that the car is hidden behind one of the other two doors (although we do not know behind which one). Most of the participants of the show did not see the benefit in switching the door, thinking that their chances to win remain the same – 33.3 percent. However, it is not true! In fact, the chances of winning the car after changing the initial choice are doubled. Yes, initial chances to win the car are the same 33.3 percent, whichever door you choose. But after opening the door with a goat, the chances that the car is hidden behind the rest, the third door, are 66.6 percent.

The easiest way to calculate these probabilities is to imagine that you are choosing between your door (33.3 percent probability) and the combined probabilities of the two remaining doors (66.6 percent respectively). After all, when you choose a door, the probability that the car is behind the other two, equals 66.6 percent. When you find a goat behind one of these doors, the possibility to find a car behind the other door remains 66.6 percent.

The importance of finding the true value

This problem shows how easy it is to fall into a trap when treating non-random data as if it were random. Modern British TV show – Deal No Deal – where are 26 boxes with various amounts of money in them, is a tribute to its American pioneer. It also effectively demonstrates how incompetent an average person when playing a game of chances is. Participants of the show do not understand their mathematical chances to win and act intuitively. This problem has a lot to do with gambling, where players make disadvantageous decisions.

Betting requires the skill to understand whether the odds offered by a bookmaker represent the probability of that event to occur. It doesn’t matter if it’s a TV show, a lottery, or online sports betting – understanding and finding the true value, is the key to make profit.

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