What is a Monty Hall problem?
The Monty Hall problem is a probability puzzle based on the American television game show Let's Make a Deal. The problem is named after the show's original host, Monty Hall. The problem is as follows: a contestant is presented with three doors, behind one of which is a prize. The contestant is asked to choose one of the doors. After the contestant has chosen a door, the host (Monty Hall) opens one of the remaining doors, revealing a goat. The contestant is then asked if they would like to switch their choice to the other unopened door. The question is whether or not the contestant should switch their choice in order to have a higher chance of winning the prize.
The Monty Hall problem is a probability puzzle named after the host of the television game show "Let's Make a Deal," Monty Hall. The problem goes as follows:
Suppose you are a contestant on a game show, and there are three doors in front of you. Behind one of the doors, there is a valuable prize, like a car, and behind the other two doors, there are goats.
You are asked to choose one of the three doors without knowing what is behind them. Once you have made your choice, the host, Monty Hall, who knows what is behind each door, opens one of the other two doors to reveal a goat. Now, there are two doors left—one that you initially chose and one that remains closed.
At this point, Monty gives you a choice: you can either stick with your original choice or switch to the remaining unopened door. The question is, should you stick with your initial choice, switch doors, or does it not make a difference?
The surprising answer is that your chances of winning the car are higher if you switch doors. The probability of winning by switching is 2/3, while the probability of winning by sticking with your original choice is only 1/3.
To understand why this is the case, we can think of the probabilities after the host opens one of the doors. Initially, there was a 1/3 chance of picking the car and a 2/3 chance of picking a goat. When Monty reveals one of the goats, the probability that the car is behind the unopened door you didn't choose increases to 2/3. Therefore, switching doors gives you a higher chance of winning.
This solution might seem counterintuitive since there are now only two doors left, but it can be proven mathematically using conditional probability. The Monty Hall problem is a fascinating demonstration of probability and often confounds our intuition.
The Monty Hall problem is a probability puzzle named after the game show host Monty Hall. Here is a step-by-step explanation of the problem:
1. The game begins with three doors. Behind one door is a valuable prize, while the other two doors hide goats (or some other undesirable outcome).
2. The contestant chooses one of the doors, let's say Door 1, without knowing what is behind it.
3. After the contestant chooses, the host, who knows what is behind each door, opens one of the remaining doors that does not hide the prize. Let's assume the host opens Door 3, revealing a goat.
4. At this point, the contestant is faced with a new decision. They can either stick with their original choice (Door 1) or switch to the other unopened door (Door 2).
5. The question is: Should the contestant stick with their initial choice or switch to the other door to maximize their chances of winning the prize?
The answer, which might seem counterintuitive, is that the contestant should always switch to the other unopened door. By doing so, they increase their chances of winning the prize from 1/3 to 2/3. This result is derived from the fact that the host's action of revealing a goat provides additional information that changes the probabilities.