The only reason people get confused about the Monty Hall problem is that the problem description rarely if ever makes it clear that the host knows where the car is and deliberately chooses a different door.
It's inconceivable (for example) that Paul Erdos, a world class mathematician, would fail to solve this problem if it were actually communicated clearly.
It is incredibly annoying that in the case where the host doesn't know where the car is but opens a goat door anyway, the probability goes back to 50-50
Original rules (host knows where car is and always opens a door with a goat):
- 1/3 of the time your original choice is the car, and you should stick
- 2/3 of the time your original choice is a goat, and you should switch
Alternative rules (host doesn't know where car is, and may open either the door with the car or a door with a goat)
- 1/3 of the time your original choice is the car, the host opens a door with a goat, and you should stick
- 1/3 of the time your original choice is a goat, the host opens a door with a goat, and you should switch
- 1/3 of the time your original choice is a goat, the host opens the door with the car, and you're going to lose whether you stick or switch
So even under the new rules, you still only win 1/3 of the time by consistently sticking. You're just no longer guaranteed that you can win in any given game.
Well yes, if you throw out half of the instances where your original choice was wrong, then the chance your original choice was correct will inevitably go up.
That would indeed be annoying, but I doubt it is the case. If you only consider this scenario, it cannot be distinguished by conditional probability from the case that the host knows, and so the math should stay the same.
As usual, the problem is not an incredibly difficult problem, but just a failure to state the problem clearly and correctly.
Try to write a computer program that approximates the probability, and you'll see what I mean.
Your program shows exactly what I mean: "Impossible" cannot be non-zero, your modified question is not well-defined.
Yes, of course it depends on the host knowing where the goat is, because if he doesn't, the scenario is not well-defined anymore. This is not annoying, this is to be expected (pun intended).
The scenario is well-defined. There's nothing logically impossible about the host not knowing which door has the car, and still opening the goat door.
"Impossible" in the program just refers to cases where the host picks the car door, i.e. the path that we are not on, by the nature of the statement. Feel free to replace the word "impossible" with "ignored" or "conditioned out". The math remains the same.
No, sorry, it is not well-defined. But I should have been clearer. What is not well-defined? Well, the game you are playing. And, without a game, what mathematical question are you even asking?
You cannot just "ignore" or "condition out" the case that there is a car behind the opened door, the game doesn't make any sense anymore then, and what you are measuring then makes no sense anymore with respect to the game. In order to make it well-defined, you need to answer the question what happens in the game when the door with the car is opened.
You can for example play the following game: The contestant picks a door, the host opens one of the other doors, and now the contestant can pick again one of the three doors. If there is a car behind the door the contestant picks, the contestant wins. Note that in this game, the contestant may very well pick the open door. The strategy is now to obviously pick the open door if there is a car behind it, and switch doors if it is not. I am pretty sure, when you simulate this game, you will see that it doesn't matter if the host knows where the car is (and uses this knowledge in an adversarial manner), or not.
The game you seem to want to play instead goes as follows: If the door with the car is opened, the game stops, and nobody wins or loses. Let's call this outcome a draw, and forget about how many times we had a draw in our stats. But you can see now that this is an entirely different game, and it is not strange that the resulting stats are different than for the original game.
It's inconceivable (for example) that Paul Erdos, a world class mathematician, would fail to solve this problem if it were actually communicated clearly.