[pgt]

The Monty Hall problem is typically not clearly explained with the confounding precondition that the game show host won’t open a door that reveals the prize. Once I understood that there was a bias against one of the doors, the problem became obvious.

[mannykannot]

This is a fairly common reacton to the puzzle these days.

In the time and place of the problem's first widely-distributed posing (Parade Magazine, 1990) it might reasonably be assumed that many people seeing it there were somewhat familiar with the game show it is loosely based on.

Even without that background, one might reasonably deduce that the second door opening never reveals the prize, as there would be no point, in that case, in asking whether the contestant wants to change her pick.

That is a somewhat meta argument, in that it involves understanding human intent and what makes for a good puzzle, and it means that it is not just a math/logic/probability challenge, but it is nevertheless a good puzzle. There's no law that says a puzzle must explicitly state every fact of the matter.

[mjburgess]

I agree. I think a lot of these probability puzzles end up being "illusions of emphasis", ie., that we fail to grasp the problem because how components of it are expressed in language.

When the host opens the door and why need to be repeated several times and with lots of emphasis before asking the question. These ordinarily minor details are extraordinarily relevant, and the language used should express that.

[Rerarom]

Yes, because if the problem was expressed purely formally from the start, there could not possibly be any controversy. The appeal of Monty Hall is precisely this interplay of natural and formal language.

[daveFNbuck]

People aren't generally very good at probability. Plenty of people get it wrong despite understanding the puzzle correctly.

[kragen]

It doesn't matter if there's a bias against one of the doors if you're only considering the cases where the door Monty opened did happen to reveal a goat. In two thirds of those cases, you can improve your choice by switching. It doesn't matter if Monty knew the goat was there ahead of time or not!

I was going to make a lengthy logical argument, but I started to wonder if maybe I was reasoning incorrectly as I tried to prune the redundant cases, so instead I took two minutes to write this Python program to enumerate all 27 equally probable possibilities; if you pipe its output to sort | uniq -c you will see that I am correct:

    xs = [(you, monty, car) for you in range(3)
                            for monty in range(3)
                            for car in range(3)]
    print(xs)
    for you, monty, car in xs:
        if monty == car:
            print("Monty revealed the car, too bad")
        elif you == car:
            print("You win only if you do not switch")
        else:
            print("You win if you switch")

[kragen]

Now I realize this is dumb because I wasn't excluding the cases where Monty opens the door you picked. Excluding those cases takes us back to 50:50! So I guess it really does matter that Monty knows in advance where the goat is...

How embarrassing that it took me two minutes to get the wrong answer and two hours to realize it — just long enough that I couldn't delete my confident assertions above :) This probably forms some kind of evidence about the quality of evidence you can get out of computer experiments without careful thinking :)

[analog31]

I assumed that the game -- at least the math problem -- starts when Monty has opened a door and revealed a goat.

[jonsen]

The point is if Monty knows he’s going to reveal a goat or if he just does so by chance.