I'm having trouble understanding something about the monty hall problem. If monty opened one door before you arrives, then you would have a 50/50 chance, whichever door you picked, because there are only 2 doors to pick from - right?
So, what about if you arrive, and monty said "I assumed you would pick door 1, so I opened door 3"? In that case, picking door 2 would be the same as if you had picked door 1, and then switched, so you should have 2/3 odds on it. But if monty hadn't said anything and you just picked one of the two doors, then your chances would be 50/50... or would they? Does monty assuming that you would want to pick door 1, change the probabilities even if you don't know about his decision?
What about if someone else picks a door and tells monty, who opens another one, and you then have the option to choose - without knowing which door the original person picked? What are the probabilities then? Does one of the doors secretly have a 2/3 probability? That doesn't make any sense to me.
Can someone explain this? Because I'm really not getting it.