:smack: OK, sorry for double posting, but my previous post probably came out sounding too harsh (and a bit stupid). I was just trying to make a joke.
I’m not sure that’s the case here. Monty seems harder for people to grasp when they know more math, not less.
I think the issue is it ‘feels’ like it’s a ‘gambler’s dilemma’ type problem, so people are fighting against how they taught themselves to deal with probability from the very beginning. Once you get past that part and reason out what the actual question was (Did you guess right the first time?) it’s easier.
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Dude ran an LEGENDARY D&D campagin back in the 80s, though. People still talk about it.
OK, I was suddenly unsure of exactly when the switch was offered in the Mythbusters episode, so I had another look to make sure. As I thought (and as it should be), first (1) the contestant selects a door, then (2) an empty door is revealed, and only then (3) the chance to switch to the remaining door is offered.
No. In the episode, at the point where a contestant is given the choice to switch, a door has been opened and shown to be empty. At this point, random or not (that is, the host knew that the door he opened was empty versus it could have had a price behind it) makes absolutely no difference. Once it is show to be empty, you get a 2/3 chance to win if you switch.
But you weren’t, so this makes no difference. Again, at the point when the switch was offered, a door had already been opened and show to be empty. At this point (at the risk of sounding like a broken record) you get a 2/3 to win if you switch.
And yet, all 20 contestants stuck to their original door.
Leaving aside the classic problem as being discussed here, does anyone know whether on the actual TV show Monty Hall always offered the switch? I must say, not having seen the show, my assumption would have been that he may have been playing non-random psychological games (and offered/not offered the switch accordingly) in which case the Monty Hall The TV Show would have very different strategy than Monty Hall the Math Problem.
What I learned from this episode was how much of a factor post hoc rationalization is.
I think if you asked people what are the chances of picking the right door out of three choices, most of them could correctly tell you they’d have a 33% chance. But once they’ve committed to a door, they somehow convince themselves that if was the right choice.
I’d be interested in seeing this psychology explored in experiments. Suppose you ran the experiment I described earlier where the contestant picked a door and was then given the choice of trading it for both of the other two doors. Would people still feel that they should stay with their first choice? Or suppose somebody else picked the door for the player and Monty then made them the standard offer. Would people feel as committed to their first door if they hadn’t chosen it?
[QUOTE=Wikipedia]
Because of his work on Let’s Make a Deal, Hall’s name is used in a probability puzzle known as “The Monty Hall Problem,” which examines the counter-intuitive effect of switching one’s choice of doors, one of which hides a prize, if “Monty” reveals an unwanted item behind a door the player did not choose. Hall himself gave an explanation of the solution to that problem, and why the solution did not apply to the case of the actual show, in an interview with The New York Times reporter John Tierney in 1991.[9] Because Hall had control over the way the game progressed, he played on the psychology of the contestant.
[/QUOTE]
So yeah, it seems that the actual show Let’s Make a Deal could play out very differently from the puzzle. Which may be one reason why everybody seems to have this urge to over-complicate the bejeezus out of things when talking about the strictly defined math problem.
I believe you are wrong here. But let me back up and start from the beginning.
So, slightly different rules. Again, one prize, three doors. Again, you select a door. Then, the host randomly opens one of the other two doors, not knowing where the prize is. If the prize is revealed in this fashion, game over, you lose. If the prize is NOT revealed, you are now given the option to switch. In this modified version, then switching is in fact 50/50 no better than staying. Why?
So, you do this 99 times. 33 times, you guessed right to begin with. In all 33 of those, an empty door is revealed, then you should NOT switch. 33 times, you guessed wrong, and then the prize was revealed, and you lose. 33 times, you guessed wrong, and then the prize was NOT revealed, and you SHOULD switch.
So out of 99 playings, 66 of them proceed far enough to get you a choice, and in those 66, 33 of them are you-should-switch and 33 are you-should-not, so it’s 50/50.
The point I’m trying to make, however, is that the way the problem is usually (or at least, often) phrased does NOT in fact truly specify the rules… that is, it just specifies a single instance: “You are on a game show, there are three doors, prize behind one, you choose one, host opens another one, reveals it to be empty, gives you the chance to switch”. Purely from that description, we can not definitely distinguish whether this is the “real” monty hall problem or not. For instance, it could be one instance of the random-reveal game discussed above.
In that case, I think there’s not so much the “ooh, I must have been right initially” factor as the “this guy is giving me an offer that seems ridiculously good, he must know something I don’t”. Which of course gets back to the “are the rules that x will always happen, or is this just a single playing of the game” issue.
Now I’m confused. I thought Carol Merrill opened the doors, not Monty.
As far as the psychology of switching vs. not switching, I think a lot of it has to do with the feeling that if you switch and you lose then you feel extra bad because your first choice turns out to have been the right choice, whereas if you don’t switch and you lose then you can rationalize it because you chose wrong in the first place.
In other words, if you switch and you lose then it’s like you were going to win but you lost by switching. If you don’t switch and you lose then it’s like you were not going to win anyway, which doesn’t feel so bad.
No, I still think I’m right. Here’s why: You pick a door. OK, you now have a 1/3 chance of having picked the one with the prize behind it. Monty gets the two other doors. His set of two doors has a 2/3 chance of containing the prize.
Then Monty flips a coin as to which as to which of his doors he opens. If he opens one and it has the prize, then you lose, game over. Fair enough. If it doesn’t have the prize, however, you get a chance to switch.
*Nothing about this changes the original probabilities. *Your door still still has a 1/3 chance, Monty’s two doors still has a 2/3 chance, and the offer to switch is still equivalent to swapping your one door for Monty’s full set of two doors.
Your ability to pick the right door to begin with doesn’t suddenly become better depending on Monty’s door selection technique.
Because of the 2/3 chance of Monty’s doors having the prize, half of the time the prize is shown and you lose no matter what. This means you don’t increase your chances by swapping, since half of the time you would have benefited by swapping now won’t. This results in equal chance of winning by swapping when you have the choice (1/3 win by swap, 1/3 lose no matter what, 1/3 lose by swap vs 2/3 win by swap 1/3 lose by swap in the original)
I’m an idiot. That sounds about right. :smack:
Edit: Hang on. If I’m an idiot… did the contestants on Mythbusters know whether the host’s choice of door was random or not? I’m not sure if I can be bothered to go back and check. That would make a difference, though, right?
As I recall, Adam said the rules of the game were explained to the players. But he didn’t specify how in depth the explanation was.
Maybe that’s true in most situations but would it be a person’s first instinct in a game show? Monty isn’t paying for the prizes so he has no reason to be stingy. Presumably an episode was more popular when people won big prizes.
Right, and is exactly the point I’ve been trying to make in my last several posts. Simply presenting one incident of the game is not the same as presenting the game with the rules that are necessary for it to be the “Monty Hall Problem”.
Got any sauce I can put on this humble pie?
I recall Adam explaining the rules direct to camera, but I don’t recall him saying that the contestants were told the condition.
Thanks for bring this up, friedo. First off, I’d like to point out that Cecil completely dropped the ball twice, the first by not reading the question carefully enough in the first place, the second, in what I can assume was a pathetic attempt at a cover-up, by inventing that arglebargle about Monty not being stupid. Dude, that wasn’t part of the original question. You completely made that up. If you think the original, simple problem isn’t worthy of a full column, solve it, then post your follow-up question. You’ve done this many times. This is a textbook case of how a complete inability to admit that you’re wrong can bite you.
Anyway, glad Mythbusters brought up the psychological aspect. As my humble offering to this thread, let me bring up a situation that might make it a little easier to understand.
The game: Deal or No Deal. You’ve gone the distance…a pretty impressive feat in itself…and only two cases remain. One of them contains $750,000, the other some modest figure, say, $1,000*. The Banker, of course, made one last pathetic lowball offer, but you shut him down colder than Antarctica. As is the custom, you have the option of opening the case that’s been by your side the whole game, or the one still in the hands of a pretty spokesmodel up on the stage, and whichever figure you reveal, that’s your winnings.
Which one should you choose? Well, one of them has the big prize, the other doesn’t, and you have absolutely no information which can help your decision (which is not the case in the Monty Hall problem). So it’s a 50/50, feast or famine, lady or tiger, Heaven or Hell pick either way.
But suddenly, a friend on the sideline calls you over and shows you a photograph. It was taken by a recently-fired studio employee earlier today. (You have no idea how it wound up in your friend’s hands, but curiosity can wait.) It’s a bombshell…the $750,000 plate being placed in the case that’s up on the stage!
Now, you can’t be absolutely sure that it’s still in there. There’s a chance that photo was leaked somehow and the plate got moved, or the case is going to be used in a later episode. But you figure the chances of that are pretty small. In all, the smart move…the sane move…is to take the case on the stage.
The show returns from commercial, and Howie Mandel finally asks the fateful question…
"The time has come to decide. You can either open YOUR CASE, or switch YOUR CASE for the case that’s on the stage. If you open YOUR CASE, then the amount in YOUR CASE is what you walk away with; otherwise, you open the case that’s not YOUR CASE and get the amount in THAT OTHER CASE, and not YOUR CASE. If you wish to open YOUR CASE, open YOUR CASE now, otherwise give YOUR CASE to me, I’ll get THAT OTHER CASE, which is most definitely not YOUR CASE, but it will have to serve as YOUR CASE because you didn’t open YOUR CASE.
Now bear in mind, there was absolutely nothing special about “your” case. It was a case you decided, for whatever reason, you weren’t going to eliminate, nothing more. Functionally, that makes it no different from the case on the stage! And yet, from the tone of Mandel’s voice…to say nothing of the roughly 300 times he said “your case” over the course of the game…the implication is that you’d be doing something very iffy, at best, if you switch.
And all of a sudden, what was supposed to be a no-brainer suddenly turns into the most gut-wrenching decision of your life, and you wonder why you couldn’t at least tried to get on Jeopardy. Come to think of it, what the heck was the point of casting a completely neutral figure like the Banker as a villain, anyway…
(I’ve seen several players go the distance, and not one even considered switching.)
- Given the time it takes to get on this show, and the fact that each contestant gets only one chance, ever, you can take home a lot more than one cent and still leave bitterly disappointed. Really don’t see how that particular figure is such a big deal.
Okay, can someone explain this variation?
Standing in a hallway,I ask you to choose “left or right.” we wait a few minutes then I bring you into a room with two doors with one prize randomly placed behind one of two door; you are allowed to look behind the door on the left or right based on your choice before you entered the room. Question A: do you agree that the odds that the door you open has a prize is 50%?
Now, suppose I tell you that after you chose “left or right,” Someone did the Monty hall setup with 3 doors and a guy name Joe inside the room we were waiting outside. After Joe chose, the other host opens a remaining door with no prize. The Host then removes that door from the stage. Now Joe is standing in front of two doors. According to the view presented in this thread and by mythbusters, The door joe chose has 33% chance of being the door with the prize, and the remaining door he did not choose has a 66% chance of having the prise. At least, this what I believe many are arguing here (including the mythbusters and other web sources). Do I have this striaght?
So joe leaves the room with one door having double the odds of having the prize than the other.
BUT, if you answered “YES” to my question A above, as soon as you walk into the room, the very same set of doors immediately have different odds of hiding a prize??? Or better yet, while you are standing outside the room and Joe is standing in front of the two remaining doors, do the doors have both equal and unequal odds of hiding the prize depending on who’s perspective?!
Does this sound like quantum physics at a macro scale??? Knowledge and observation changes the the state of some kind of probability wave/particle know as the game-show-itron?
Is my setup flawed? Is Schrodinger’s cat the prise… is it dead?
I’m confused.