I don't think you need to take data.
Don't you think we would have remembered if Monty had ever screwed the contestant, and showed the Grand Prize as the first door the contestant didn't win?
"Ok. Let's look at one of the doors you didn't select. ....... You didn't select the Grand Prize! Hurrah! So, do you want to keep your loser door, or take the other loser door?"
Whatever Monty may or may not have know about probability theory, he surely knew that doing that wouldn't make for an entertaining climax to his show. As long as he was committed and able to avoid that scenario, the usual analysis is fine.
Even if Monty was feeling bitter one day, and decided to screw the contestant, or they just made a mistake and opened the wrong door, that really isn't the point of the problem.
Of course your best strategy is different for Screw You Monty than for Let's Have an Exciting Climax to the Show Monty, but that shouldn't surprise any of us. Any optimal strategy has to assume a prior on the behavior of Monty. No Free Lunch.
Don't you think we would have remembered if Monty had ever screwed the contestant, and showed the Grand Prize as the first door the contestant didn't win?
"Ok. Let's look at one of the doors you didn't select. ....... You didn't select the Grand Prize! Hurrah! So, do you want to keep your loser door, or take the other loser door?"
My understanding was that Monty Hall knew which door contained the prize. After he revealed a non-prize door, he more strongly encourage players to move when their initial guess was correct.
In class number four of the online Artificial Intelligence course from Stanford, Peter Norvig explains "one of the most popular problems in the subject of probability theory, the Monty Hall problem". His presentation was exactly what I have always been taught, and I got the right answer on the quiz.
Hold; P(goat) = 2/3; P(car) = 1/3;
Switch; P(goat) = 1/3; P(car) = 2/3.
Since the last time I looked at that wikipedia article, a number of qualifications and stipulations and elaborations have been added. Apparently this problem has kept its popularity and controversy with undiminished vitality since around 20 years ago, when it was popularized by a national Sunday newspaper magazine columnist Marilyn Vos Savant. Her answer is the one above, which is also what I would call the least complicated wikipedia page answer (as the page sits today.)
Now there is one probable error in Norvig's lecture. He claimed that Monty Hall didn't know the answer to his own problem, and quotes a letter he says was written by Monty Hall in 1990. If this is true, I find it impossible to follow the following New York Times story, from 1991 where Monty explains all the subtleties of the problem in much shorter order than all of the complications on wikipedia. To summarize Monty's take: he is the carnival barker; it's his game; and he can change the rules in the middle of the game if he feels like making it more complicated than your five minute presentation for your students.
But there is one critical element missing from all of the discussions I have seen to date, except for one which was told to me by a playfully deviant fellow many years ago. His argument is simple: if by this strategy a player can double his chances of winning, the players would have figured this out forthwith; everybody would have switched every time; and the drama and the fun would have been drained from the game. The fact is the show was the most popular game show in it's hay day. Ergo there cannot possibly be an easy way to game this strategy. My deviant probability theorist pal had confidence in Monty's game players to find any available edge.
Also the data is in the archives. All you have to do is watch every episode and count the holds and switches and wins and losses and you will have a really high confidence estimation whether the correct answer is closer to 2/3 and 1/3 or is closer to .5 -- .5. If no statistician is interested in researching this, surely some enterprising historian of science or sociologist of science ought to be up for it.