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.
This is from an era when you could still make the computer do what you want. Those days are long gone now; incidentally, that's why "It is now safe to turn off your computer" does not belong in this list, because it still exists, just not under that name. Windows (or whatever OS) decides when you can turn it off. Just like it now decides when it's time to upgrade.
Actually there _is_ a lot to be missed about those times, in spite of all the "progress" we've made since then.
"It is now safe to turn off your computer" was a kludge for running Windows 95 on PCs with a physical on/off switch (as opposed to newer ATX power supplies, which have a pushbutton to turn them on and can be switched off via software). I guess it's theoretically possible to build a modern PC with a 30+ year PSU, and then you would see this screen again - or, more realistically, you could disconnect the PS_ON line (https://electronics.stackexchange.com/questions/394414/how-d...) from the mainboard, preventing the system from turning off the PSU.
My 3D printer has a physical off switch. And it runs Linux - debian - on an embedded ARM board, so I can ssh into the printer etc. (I can even hook up a keyboard and HDMI monitor)
I fear about my filesystem every time I notice the printer is running at night (after a long print job) and I just turn it off without going over to my pc, ssh in and shut down the OS.
Still, it hasn't eaten my filesystem, yet ... ext4 journaling DOES seem to work.
It's technical jargon. Scientists used speak a mishmash of Greek and Latin, since those were common among educated people. That hung on for longer than was realistic, but it became a kind of slang for group recognition.
This department was named in the 1880s. They probably would just call it the Apple Department today, though I admit I still like the tone that sort-of-Greek-or-Latin rings.
