[ALittleLight]

"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.

[antris]

Strictly speaking it's not the hosts knowledge that changes the problem, but the fact that as a contestant you know how the host operates. This makes you able to extract information about the choice the host makes.

Different ways of host operation changing the outcomes are listed here: https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_...

>"Monty Fall" or "Ignorant Monty": The host does not know what lies behind the doors, and opens one at random that happens not to reveal the car. Switching wins the car half of the time.

[tgb]

You stimulated out the situation where Monty Hall always picks the door without the prize. That’s exactly the standard Monty Hall problem. Change your code to instead allow him to choose the prize door (but never the player’s door). Then condition on him picking the goat door by dropping all the cases where he picked the prize door.

You’ll see that that changes it by discarding scenarios where switching is good (the prize is shown to you) but not ones where staying is bad.

[v-erne]

I always believed that saying that Monty knowledge affects the odds is just more people friendly way of saying that those are not independet events and simple probabilistic model does not work when Monty is not allowed to select prize when revealing gate - You need something fancy (like a bit of Bayes conditional probability)