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The way I like to think about it is as follows: You pick door 1. Out of three doors you have a 1/3 chance of having a car behind your door. The chance that the
by zaksoup 12y ago
The way I like to think about it is as follows: You pick door 1. Out of three doors you have a 1/3 chance of having a car behind your door. The chance that the car is behind one of either of the other two doors is 2/3. Let's pretend that instead of opening a door and revealing a goat, Monty instead says to you "You can switch to both of the remaining doors. If the car is behind either one of them you get to keep it." Your likelyhood of getting the car by switching is now 2/3.
Instead, Monty does the statistical equivalent: He allows you to know for sure which of the other doors definitely has a goat.
- choffstein 12y agoThis is a great answer. To answer the question about the 3-sided die: the question is flawed for 2 reasons. First, the way you phrased it, you know the information a priori. It's like me learning that the car isn't behind door #1 before I choose. In that case, I have a 50/50 chance on the other doors. But what if you didn't tell me before? This introduces our second problem. Let's say I roll the die and you hide it under a cup. I tell you I chose side A and you tell me A is physically impossible and ask me if I want to switch. Well of course I want to switch ... but I didn't learn any more information about sides B and C. It's a 50/50 toss-up. In terms of the game show, you basically just opened the constant's door. A critical element of the original problem is when the information is learned and knowing that the contestant's door will not be opened. It is what allows the phrasing from zaksoup above.
- Duhveed 12y agoThank you for a great answer. I've known about this problem for a long time and never had a way to intuitively think about it. I will now be stealing yours. :)