Monty Hall Problem

The basic thing to keep in mind is that initially, your chance of choosing the door with the prize is 1 in 3. This does not change when Monty shows you (as he must, under the rules) that one of the remaining doors has a goat. You knew that already, you just didn't know which door. So, if your door doesn't have the prize, then the one he didn't reveal must have the prize. Since there is a 1 in 3 chance that your door has the prize, there must be a 2 in 3 chance that the one he didn't open has the prize, and it pays to switch.

One way to think about it is to suppose that there are 100 doors, one of which has the prize. The chance that you initially choose the door with the prize is only 1 in 100. So, there is a 99 in 100 chance that one of those other doors has it. Monty knows which one it is, so he can open 98 doors and never reveal the prize. He will do this in 99 out of 100 games played, and in those 99 out of 100 games, the door he doesn't open will have the prize. In the 1 out of 100 games played where you chose the prize door, he'll open 99 doors at random, and the remaining door will have a goat. So in 99 out of 100 games played, it pays to switch.

I left the class with several variations of the problem to think about for class on Friday:

1. Ignorant Monty: Monty doesn't know where the prize is, and sometimes randomly opens the door with the prize. Is it an advantage to switch, to stay, or doesn't it matter?
   
2. Angelic Monty: Monty knows where the prize is. If you choose the door with the prize, he opens it and congratulates you. If you choose the wrong door, he opens a door without the prize and offers you the chance to switch. Is it an advantage to switch, to stay, or doesn't it matter?
3. Monty From Hell: Monty knows where the prize is. If you choose the door with the prize, he opens a door without the prize and offers you the chance to switch; but if you choose a door with the goat, he opens it and says, "too bad, you lose." Is it an advantage to switch, to stay, or doesn't it matter?
4. I didn't mention this, but we'll think about it as well. This is "Mixture Monty." Before coming on stage, Monty flips a coin. If it comes up heads, he behaves as Angelic Monty on stage. If it comes up tails, he behaves as Monty From Hell. Is it an advantage to switch, to stay, or doesn't it matter?