The Monty Hall Problem
Scenario
There are three doors before you, behind one lies treasure, behind the other's lie dead-ends. After you choose a door one of the dead-ends is revealed to you. You are then given an opportunity to switch to the other door. Is it better to stick to your original choice, or switch?
Play (click a door)
Switch Wins:
Keep Wins:
Answer
You have a 1/3 chance of winning if you choose not to switch.
You have a 2/3 chance of winning if you choose to switch.
Intutive Explanation
If you play the game and decide to not switch, I think it's pretty clear to see you have a 1/3 chance of picking correctly, and thus a 2/3 chance of picking incorrectly.
To understand what happens when you switch, let's assume you're playing and your intial pick is incorrect. This means if you decide to switch you get the correct answer, and remember you have a 2/3 chance of intially being incorrect. This is why you have a 2/3 chance of winning if you decide to switch, because you have a 2/3 chance of intially being wrong, but if you switch you are guaranteed to pick the correct door.
Mathematical Explanation
Baye's Theorem
\(P(A \mid B) = \dfrac{P(B \mid A) \cdot P(A)}{P(B)} \)
A and B represent events, and \(\mid \) means given. So \(P(A
\mid B) \) means the probability event A happens, given event B
happens.
Proof
Let's label the doors A, B, and C. Now let's say we pick door A, and door C is revealed to be a dead-end. What we want to find is the probability you win if you switch doors, which is: