If you’re a fan of game shows, you’ve probably heard of the Monty Hall problem. The Monty Hall problem is a famous probability puzzle based on a game show called Let’s Make a Deal, which was hosted by Monty Hall.

The game involves three doors, behind one of which is a prize, and the other two hide goats. The contestant chooses one door, and then the host opens one of the other two doors to reveal a goat. The contestant is then given the option to switch their choice to the remaining unopened door or stick with their original choice.

## The Problem

The question that arises is should the contestant switch doors Intuitively, it might seem that there’s no advantage in switching because there are only two doors left to choose from, and neither seems more likely than the other to conceal the prize. However, this intuition is incorrect.

## The Solution

The solution to this problem lies in understanding probability. When the contestant initially chooses one door out of three, they have a 1/3 chance of choosing the prize door and a 2/3 chance of choosing a goat door. After one goat door has been revealed by Monty Hall, there are now only two doors left: one with the prize and one with a goat.

When given the option to switch their choice, it’s important to note that while there are still only two doors left, there has been new information revealed by opening one of them. Specifically, we know that at least one of those doors hides a goat. This means that if the contestant sticks with their initial choice, they have a 1/3 chance of winning and if they switch they have a 2/3 chance.

## Monty Hall as Game Theory

The Monty Hall problem is often cited as an example of game theory, which is the study of strategic decision making. Game theory helps us understand how rational individuals make decisions when they interact with each other. In the Monty Hall problem, the contestant is faced with a strategic decision: whether to switch or stick with their initial choice.

Game theory provides a framework for analyzing this type of decision making and predicting the outcome. In this case, the optimal strategy for the contestant is to always switch doors since it increases their chances of winning from 1/3 to 2/3.

### Conclusion

In conclusion, while it may seem counterintuitive, switching doors in the Monty Hall problem results in a higher probability of winning than sticking with your initial choice. This puzzle has become famous not only for its counterintuitive solution but also for its application to game theory and strategic decision making.