You said: "I can make sense of it by drawing out the probability trees and see that it's 1/2 rather than 2/3 in this case"
(Edit: sorry that was not your statement, but that from furyofantares.)
Now you agreed that it is still 2/3, so I still stand by my statement "As long as the host doesn't accidentally open the prize door, it doesn't matter whether he forgot or not."
> Well, it does matter unless you have the rather unusual rule that you win if the host reveals the prize
If you change the game and say the host is blind, you have to also make a rule about what happens if the blind host reveals the prize. Otherwise how could you decide on a strategy as a guest?
> Well, there is no decision to be made if the host revealed the prize
No chance for the guest to switch, yes, but it is still a branch in the whole probability tree, and you have to count it either as a win or a loss for the guest. They can't just all go home and pretend the show didn't happen.
> Otherwise could equally well say that you have to include the probability of getting through to the final round, the probability of getting onto the gameshow in the first place, ...
The problem statement is: "What strategy (stick or switch) is best for the guest (= has higher probability to win), provided he gets into the game show final." No ambiguity here.
> If you change the game and say the host is blind, you have to also make a rule about what happens if the blind host reveals the prize.
The most natural rule is that the guest loses: the guest picked a door that didn't have the prize, they get what's behind "their" door. (And note that the much-argued formulation already "changes" the game from what was done on the original gameshow).
> The problem statement is: "What strategy (stick or switch) is best for the guest (= has higher probability to win), provided he gets into the game show final."
Why "provided he gets into the gameshow final" and not "provided he gets into the gameshow final and is offered the chance to switch"? Like, if you're asking whether a chess player should take an en passant capture, you'd look at whether capturing or declining en passant leads to winning more often, you wouldn't look at all possible chess games (including those where there was no chance to capture en passant) because that just adds a bunch of irrelevant cases.
I don't want to argue which rule would be more natural or make a more interesting show. But without a clear rule, I cannot decide on a playing strategy.
> Why "provided he gets into the gameshow final"?
Because I assumed that the host always has to open a door and give a choice to switch.
> I don't want to argue which rule would be more natural or make a more interesting show. But without a clear rule, I cannot decide on a playing strategy.
It doesn't make any difference to your strategy! You don't get any choice if the host opens the door and reveals the prize, so your strategy can't possibly be affected by what the payoff for that scenario is - even if you win 10x the prize if that happens, or get executed if that happens, it makes no difference to whether you should switch or not in the scenario where you actually do get given a choice.
No it doesn't. If the host chose randomly and revealed a non-prize, i.e. the point where you're making the choice, your odds are 1/2 for either door and your strategy doesn't matter.
(Edit: sorry that was not your statement, but that from furyofantares.)
Now you agreed that it is still 2/3, so I still stand by my statement "As long as the host doesn't accidentally open the prize door, it doesn't matter whether he forgot or not."
> Well, it does matter unless you have the rather unusual rule that you win if the host reveals the prize
If you change the game and say the host is blind, you have to also make a rule about what happens if the blind host reveals the prize. Otherwise how could you decide on a strategy as a guest?
> Well, there is no decision to be made if the host revealed the prize
No chance for the guest to switch, yes, but it is still a branch in the whole probability tree, and you have to count it either as a win or a loss for the guest. They can't just all go home and pretend the show didn't happen.
> Otherwise could equally well say that you have to include the probability of getting through to the final round, the probability of getting onto the gameshow in the first place, ...
The problem statement is: "What strategy (stick or switch) is best for the guest (= has higher probability to win), provided he gets into the game show final." No ambiguity here.