Wisdom of Crowd+Mounty Hall Problem

Описание: The Monty Hall problem is a famous probability puzzle named after the host of the TV show "Let's Make a Deal," which was popular in the 1960s and 1970s. The problem is as follows:

Suppose you are a contestant on a game show. The host, Monty Hall, presents you with three doors. Behind one of the doors is a car, and behind the other two doors are goats. You are asked to choose one of the doors. After you make your selection, Monty, who knows what is behind each door, opens one of the other two doors to reveal a goat. Monty then asks you if you want to switch your selection to the remaining unopened door or stick with your original choice.

The question is: should you stick with your original choice, switch your selection to the other unopened door, or does it not matter?

The answer is that you should switch your selection to the other unopened door. Intuitively, this may seem counterintuitive, but the probability of winning the car if you switch is 2/3, while the probability of winning if you stick with your original choice is only 1/3.

To understand why this is the case, it helps to consider what happens if you don't switch. Suppose you choose Door 1, and Monty opens Door 2 to reveal a goat. If you stick with Door 1, you will win only if the car is behind Door 1, which happens with probability 1/3. However, if you switch to Door 3, you will win if the car is behind Door 3, which happens with probability 2/3.

In essence, by opening one of the doors to reveal a goat, Monty has effectively given you new information about which door the car is behind. This information is more likely to be helpful if you switch your selection, rather than sticking with your original choice.

The Monty Hall problem is a classic example of how counterintuitive probability can be, and it has been the subject of much debate and discussion among mathematicians and game show fans alike.