> Yeah, but it's not 50/50. I think you're focusing on the wrong part there. The issue isn't the host's knowledge, it's the question of whether he always opens a door. If he always opens a door at random, and if in this particular run through he happens to open a door with a goat, then you will still double your odds if you switch. The host doesn't need to know what's behind the door for this to work!
No, if the host always opens a door, but has no clue what's behind them, you chances will not go up by switching. That's the central clue of the Monty Hall problem, which really helped me to grok why the odds go up by switching. By opening a door, a Monty Hall who knows where everything is, is giving you information in a way that a random choice cannot do.
Let's assume you always pick door 1. Then there are 6 equiprobable possible universes:
1. Prize behind door 1, Monty opens door 2
2. Prize behind door 1, Monty opens door 3
3. Prize behind door 2, Monty opens door 2
4. Prize behind door 2, Monty opens door 3
5. Prize behind door 3, Monty opens door 2
6. Prize behind door 3, Monty opens door 3
If no prize appears behind the door he opens, we're sure we're not in universes 3 and 6. That leaves behind 4 universes, in 2 of which we'll win the prize if we switch, so a 50/50 chance.
What's the difference between this version and the normal Monty Hall? In the usual version, universes 3 and 6 never existed in the first place, and universes 1 and 2 are the same. Whether he opens the 2nd or the 3rd door depends on the protocol he uses, but is irrelevant.
The 6 universes are not equiprobable. 1+2, 3+4 and 5+6 are equiprobable (1/3 each). 1 and 2 are equiprobable (1/6 each), 3 and 6 are impossible (0 each), 4 and 5 are equiprobable (1/3 each). You should switch in 2/3 of the cases.
No, if the host always opens a door, but has no clue what's behind them, you chances will not go up by switching. That's the central clue of the Monty Hall problem, which really helped me to grok why the odds go up by switching. By opening a door, a Monty Hall who knows where everything is, is giving you information in a way that a random choice cannot do.
Let's assume you always pick door 1. Then there are 6 equiprobable possible universes:
1. Prize behind door 1, Monty opens door 2
2. Prize behind door 1, Monty opens door 3
3. Prize behind door 2, Monty opens door 2
4. Prize behind door 2, Monty opens door 3
5. Prize behind door 3, Monty opens door 2
6. Prize behind door 3, Monty opens door 3
If no prize appears behind the door he opens, we're sure we're not in universes 3 and 6. That leaves behind 4 universes, in 2 of which we'll win the prize if we switch, so a 50/50 chance.
What's the difference between this version and the normal Monty Hall? In the usual version, universes 3 and 6 never existed in the first place, and universes 1 and 2 are the same. Whether he opens the 2nd or the 3rd door depends on the protocol he uses, but is irrelevant.