> The problem as stated has the host open an incorrect door and offer the player a chance to change his choice.
Correct, in this one particular instance. You cannot conclude from this particular instance that the host always opens the door and offers a change.
> Inferring that another possible variation might exist
is totally reasonable, while denying the possibility that the host might be able to choose his actions specifically to benefit or screw you
over is an unwarranted leap.
The problem statement does not put constraints on the host. You cannot solve the problem by assuming that those constraints exist and then attack those like Diaconis who point out that those constraints don’t exist and that the thing that is unconstrained matters.
There is a genuine human language problem here, NOT A MATH PROBLEM, which accounts for two differing but self-consistent views. It is a legitimate difference in views because human language is genuinely ambiguous.
"You pick a door, the host opens another door and reveals a goat. Should you switch?"
Does this mean you are in one particular situation where the host opened a 2nd door, with a goat? Or does it mean the host always opens a 2nd door with a goat?
If the host always opens a 2nd door, showing a goat, you should switch to the third unopened door.
If all you know, is this time you picked a door, then the host revealed a goat, you don't know what to do. Maybe this host only opens goat doors after you pick the right door, in order to trick you into switching? In that case switching would be the worst thing you could do.
A host with that strategy is a special case, but special cases where a potential general solution (always switch) doesn't work, are all you need to disprove the general solution. It cannot be a general solution if their is even one special case it doesn't work.
Most people interpret the problem to mean the host always reveals goats.
But if the language isn't clear on that, then you do have a different problem, whose solution is really impossible to optimize for without some more information on general host behavior or strategies. Without that information, all you can do is flip a coin. Or always stay, or always switch. You have no means to improve your odds whatever you do.
The problem statement does put constraints on the host, by specifying that the host opened an unselected door with a goat behind it, only to ask if the player wants to change his choice. The answer to the question of whether the player should change the choice is well defined. Other variations are irrelevant since they are different problems.
Your argument is equivalent to denying that 2 + 2 = 4 is correct because the author had the option to write something other than a 2 as an operand.
Nope. The probability theory doesn't work like that. When you argue that 2+2=4 you assume 2 and 2 are known and they are not.
A=you picked the car at first
B=the host opened the door
P(A|B) can be anywhere between 0 and 1.
In your calculations you assume that P(A|B)=P(A) which is correct ONLY if A and B are independent. Independence of A and B is not in the problem statement, you invented this clause yourself.
Correct, in this one particular instance. You cannot conclude from this particular instance that the host always opens the door and offers a change.
> Inferring that another possible variation might exist
is totally reasonable, while denying the possibility that the host might be able to choose his actions specifically to benefit or screw you over is an unwarranted leap.
The problem statement does not put constraints on the host. You cannot solve the problem by assuming that those constraints exist and then attack those like Diaconis who point out that those constraints don’t exist and that the thing that is unconstrained matters.