"Knowledge" is some physical state of the brain. I don't know exactly, connections in the brain, activation of neurons, different chemicals or electrical signals - it's something, some physical actual state of the brain. How could it be possible for Monty's brain-state to have an effect on the problem?
The obvious answer is the right one - it's not possible and Monty's knowledge has no effect on the game or the strategy the player should use. Monty knowing where the prize is and opening a bad door is the same as Monty not knowing where the prize is and having randomly opened a bad door. In either case, you should switch, Monty's knowledge doesn't change things.
Another way of thinking about it - how would the participant know if Monty opened a door at random or knowingly opened a goat door? What if they thought Monty knew, but Monty had actually forgotten and just coincidentally opened a goat door? Does any of this matter? No, because Monty's knowledge doesn't effect the game or strategy.
I also wrote a quick simulation in my Javascript console which confirms what I'm claiming here.
https://pastebin.com/y5G4PE75
The "pick and stay" strategy wins about a third of the time while "pick and switch" wins about two thirds - same as the original problem. Writing the code emphasizes that it is basically the same thing - Monty coincidentally reveals a goat versus knowingly reveals.
The obvious answer is the right one - it's not possible and Monty's knowledge has no effect on the game or the strategy the player should use. Monty knowing where the prize is and opening a bad door is the same as Monty not knowing where the prize is and having randomly opened a bad door. In either case, you should switch, Monty's knowledge doesn't change things.
Another way of thinking about it - how would the participant know if Monty opened a door at random or knowingly opened a goat door? What if they thought Monty knew, but Monty had actually forgotten and just coincidentally opened a goat door? Does any of this matter? No, because Monty's knowledge doesn't effect the game or strategy.
I also wrote a quick simulation in my Javascript console which confirms what I'm claiming here.
https://pastebin.com/y5G4PE75
The "pick and stay" strategy wins about a third of the time while "pick and switch" wins about two thirds - same as the original problem. Writing the code emphasizes that it is basically the same thing - Monty coincidentally reveals a goat versus knowingly reveals.