[choffstein]

This 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.