The Monty Hall Problem
Monty Hall Problem
The Problem: You are in a game show and the host shows you three doors, saying that only one will give you the grand prize. After choosing one door, the host will open one of the two doors you did not choose. The host knows where the prize is and he would not open that door, if ever you have not chosen it. Then he would give you a chance to switch to the other door. Should you switch or stick to the door you have picked first?
The Answer: You have a better chance of winning if you switch.
The Solution: The explanation for this problem is governed by Bayes’ theorem. For easier understanding of the solution, we have the following table:
PRIZE CHOSEN SHOWN SWITCH 1 1 2 Lose 3 Lose 2 3 Win 3 Win 3 2 Win
2 Win 2 1 3 Win 3 Win 2 1 Lose 3 Lose 3 1 Win 1 Win 3 1 2 Win 2 Win 2 1 Win 1 Win 3 1 Lose 2 Lose
The first column shows behind which door the prize is.
The second shows what the chosen door is.
The third shows what door/s the host can - and will - show.
The fourth column shows what the result will be if ever you switch doors.
It can be seen that out of the 18 possible outcomes, you can get a win 12 times and only lose 6 times. We then say that the probability of winning is 2/3 - better than the 1/3 probability that you will lose.
The Problem: You are in a game show and the host shows you three doors, saying that only one will give you the grand prize. After choosing one door, the host will open one of the two doors you did not choose. The host knows where the prize is and he would not open that door, if ever you have not chosen it. Then he would give you a chance to switch to the other door. Should you switch or stick to the door you have picked first?
The Answer: You have a better chance of winning if you switch.
The Solution: The explanation for this problem is governed by Bayes’ theorem. For easier understanding of the solution, we have the following table:
PRIZE CHOSEN SHOWN SWITCH 1 1 2 Lose 3 Lose 2 3 Win 3 Win 3 2 Win
2 Win 2 1 3 Win 3 Win 2 1 Lose 3 Lose 3 1 Win 1 Win 3 1 2 Win 2 Win 2 1 Win 1 Win 3 1 Lose 2 Lose
The first column shows behind which door the prize is.
The second shows what the chosen door is.
The third shows what door/s the host can - and will - show.
The fourth column shows what the result will be if ever you switch doors.
It can be seen that out of the 18 possible outcomes, you can get a win 12 times and only lose 6 times. We then say that the probability of winning is 2/3 - better than the 1/3 probability that you will lose.