The Monty Hall paradox hinges on the fact that the host knows in advance where the car is. If your scouts don't know where the herds are, but just picked an area at random to recon, then it is a different problem, and switching makes no difference.
And I think you stumbled upon the crux of the problem. I think we have a hard time grasping the fact that Monty not just picks one of the other two doors, but also specifically guarantees that it won't be the winning door.
Without that guarantee, it makes no difference.
Additionally, there are not many situations in real life that would replicate the conditions of that guarantee. So our "intuition" is simply geared towards situations where an actor can't read our thoughts (door pre-selection) before we act them out, and in such a scenario, pre-selecting/switching selection of doors doesn't matter.
But it doesn’t depend on Monty knowing ahead of time. Monty can learn after you pick, but before he reveals (to be sure he doesn’t reveal a car).
The scouts can depart after you’ve chosen the valley, find nothing in the mountain, give that information to you, and you should still switch, even if the scouts didn’t confirm where they are.
Monty(or the scouts), knowing if your original pick was right has no impact. The critical piece is that they are forced to reveal to you a bad choice. 2/3 times it’s the only bad choice to reveal because you’ve picked the other.
>> But it doesn’t depend on Monty knowing ahead of time. Monty can learn after you pick, but before he reveals (to be sure he doesn’t reveal a car).
Right, of course. I just meant he knows ahead of opening his door, it doesn't matter if he knows when you pick your door.
The scouts did not know that the herds are not in the mountain when they chose to scout the mountains.
Monty, on the other hand, does know that the car is not behind door C, when he chooses to open door C.
The difference is that Monty will never pick the door where the car is, whereas your scouts will sometime recon a region and find the herds there.
(Unless you assume that your scouts know where the herds are, and are purposefully not telling you and scouting empty areas, which would make them terrible scouts. We can expect this sort of crafty behaviour from a game show host, sure, but not from your own scouts.)
This results in different odds. With the scouts, switching makes no difference. You can run simulations or draw a full probability tree to convince yourself of it.
If the herds might have been in the valley and don't happen to be, you learn something. Your odds rise from 1/3 to 1/2. If the scouts know where the herds are and specifically checked the valley because the herds weren't there, you learn more - your odds rise to 2/3. But not as much as if they just told you where the herds are, so we're back to it being an unlikely scenario.
Knowing that Monty could not have revealed the car (vs happened not to) has an impact.
If you are objecting to how I described that, in terms of who knows what when, then you may be right and I probably could have been more precise.
But if you are disagreeing that scouts sent to investigate an area and return an answer (which this time happens to be negative) differs materially from the original question then you are wrong.
> Knowing that Monty could not have revealed the car (vs happened not to) has an impact.
Nope, it does not. The fact that he revealed a dud is what matters. The notion of him maybe revealing a car is senseless anyway because you would just then pick the car or the game would be over.
> Nope, it does not. The fact that he revealed a dud is what matters.
That's simply wrong. This has been discussed countless times on this forum. The most effective way to proceed is for you to spend 5 minutes writing the simulation that you think will prove you right. When I was on the other side of this, that experience is what convinced me.
> The notion of him maybe revealing a car is senseless anyway because you would just then pick the car or the game would be over.
That's not a very strong objection - game shows are weird sometimes and could often apparently be easily improved.
But more importantly, it's an objection to the wrong thing. My point was that the presented "find the herds" problem differed from the game show. "That would have been a bullshit game show" is just confused.
