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Monty Hall Problem

Robin said:
But in this case the odds are still 0.667 if you switch, even if the host didn't know where the car is. The fact that you now see a goat means that if you originally guessed wrong then the remaining door contains the car. And the odds of you getting it wrong are 0.667
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The problem is that going in, a full 1/3 of the time you are going to lose the game off the bat--just because the host revealed the car instead. There's no chance to switch in those situations.

Given that a goat has been revealed, we can now restrict down to the remaining 2/3 of the times when you do get a chance to switch.

Out of those times you get to switch, which occur with only a 2/3 frequency, 1/3 of the time you will have chosen correctly. 1/3 out of 2/3 of the time, which is a 1/2 chance of having chosen correctly, given that a goat was revealed.
 
insomneac said:
But you don't know Monty's intentions at all. Therefore you don't take them into consideration.
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As the problem is stated, there is not enough information to solve it. The only way to solve it in the classic understanding of the problem is to assume that Monty must always show an empty door and offer the switch. If that's not stated explicitly, the problem can't be formally solved without some assumptions.
 
Cabbage said:
Actually, you are the one making assumptions about the problem--you're assuming he is operating in a "fair" manner, and that he always reveals a goat.

I'm saying I don't know--maybe he reveals a goat sometimes, maybe sometimes he doesn't. Maybe he's operating in a deceptive manner--Always revealing a goat and offering the switch when I'm initially right, and not revealing a goat when I'm wrong.
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GODDAM, HOW RETARDED CAN YOU BE? IF HE DOESN'T REVEAL A GOAT, IT'S STILL TO YOUR ADVANTAGE TO SWITCH.
 
CurtC said:
As the problem is stated, there is not enough information to solve it. The only way to solve it in the classic understanding of the problem is to assume that Monty must always show an empty door and offer the switch. If that's not stated explicitly, the problem can't be formally solved without some assumptions.
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Okay, you too may be a retard. The problem says that Monty opens a door. The problem does not say you do multiple trials. The problem can be solved as stated. You don't ASSUME jackcensoredbyMrsGrundysvag, and you don't care what Monty ALWAYS does. You are presented with one trial and you have to make the most advantageous possible decision. Switcheroonimo, mcvalta arooni-mo.
 
Cabbage said:
The problem is that going in, a full 1/3 of the time you are going to lose the game off the bat--just because the host revealed the car instead. There's no chance to switch in those situations.
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Now you are just making scensoredhit up.
Given that a goat has been revealed, we can now restrict down to the remaining 2/3 of the times when you do get a chance to switch.

Out of those times you get to switch, which occur with only a 2/3 frequency, 1/3 of the time you will have chosen correctly. 1/3 out of 2/3 of the time, which is a 1/2 chance of having chosen correctly, given that a goat was revealed.
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Well, son, you kin use them fancy italicks all ye want, but in this part of the country, 1/3 of 2/3 is 2/9, not 1/2..... now gowon! Squeal! Squeal lahk a pig! Did I mention you got a purty mouth?
 
TeaBag420, if I felt you were here to learn, or even teach, as you may feel the case to be, I would attempt to help clear your confusion. As your intention instead seems simply to be to ◊◊◊◊ with people (either that, or your reading comprehension just sucks), I won't waste anymore time in that attempt for your sake.
 
Robin said:
But in this case the odds are still 0.667 if you switch, even if the host didn't know where the car is. The fact that you now see a goat means that if you originally guessed wrong then the remaining door contains the car. And the odds of you getting it wrong are 0.667
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The probability that you guessed wrong was initially 2/3. But probabilities can change if you get additional relevant information. The relevant information here is that you saw a goat instead of the car. You were more likely to see a goat if you picked the car than if you didn't. (If you picked the car, you were certain to see a goat; if you didn't, you had a 50:50 chance of seeing a goat.) So, since you did in fact see a goat, that increases the probability that you picked the car. Before, the probability you picked the car was 1/3; now, it's 1/2.

(Ok, that was the Bayesian version ... now for the frequentist version ... )

It's true that 2/3 of all your guesses will be wrong. But that 2/3 includes some cases where the host subsequently reveals the car. Since we now know that in this particular case the host didn't reveal the car, we shouldn't still be counting the whole 2/3. We should only be counting that part of the 2/3 where the host doesn't reveal the car, i.e., half of it. So we're left with 1/3 where you guessed wrong, and 1/3 where you guessed right. In the cases that are consistent with the current one, you guessed right as often as you guessed wrong. Therefore, the probability is now 1/2.
 
Cabbage said:
TeaBag420, if I felt you were here to learn, or even teach, as you may feel the case to be, I would attempt to help clear your confusion. As your intention instead seems simply to be to (censored) with people (either that, or your reading comprehension just sucks), I won't waste anymore time in that attempt for your sake.
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So you REALLY think that one third of two thirds is one half? You're willing to accept the inescapable conclusion that one and one half equals two thirds?

While your use of bad language is disturbing, I am even more distressed, nay, perplexed by your ignorance of grade school arithmetic.

Thank you for your magnanimous offer to clear up my confusion, but 1/3 of 2/3 is NOT and NEVER WILL BE 1/2.
 
TeaBag420 said:
So you REALLY think that one third of two thirds is one half? You're willing to accept the inescapable conclusion that one and one half equals two thirds?

While your use of bad language is disturbing, I am even more distressed, nay, perplexed by your ignorance of grade school arithmetic.

Thank you for your magnanimous offer to clear up my confusion, but 1/3 of 2/3 is NOT and NEVER WILL BE 1/2.
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Okay, one more time (just on the offhand chance you actually do have a brain):

If you reread, you'll notice I didn't say "1/3 of 2/3"; I said "1/3 out of 2/3" (there's that reading comprehension I was referring to earlier). 1/3 out of 2/3 is 1/2. Just like 1 out 2 is 1/2. Or 3 out of 6 is 1/2. Or 1/4 out of 1/2 is 1/2. Or, in general, just like x out of 2x is 1/2 (if x is any nonzero real number).
 
hgc said:
I know most of you are familiar with this one already, but we must settle it once and for all...
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You were right; this is always fun to watch.
 
Robin said:
No, it does not matter if the contestant knows what Monty is going to do, or what Monty's intentions are. It makes no difference.

If the conditions as set out in the problem are satisfied then the answer is "Yes - you double your chances of winning by switching".

Somebody please show me a case where Monty can affect the outcome.
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Easy. If Monty only offers the switch when you have chosen the car, then you never win by switching.

This is a legitimate criticism. The probability calculations only work if Monty offers the choice indiscriminately.
 
gnome said:


I'm not sure where you get this. [/B]
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gnome,

There are three doors on the stage. Only one of them has the real prize. The probability you got it wrong the first time is, therefore, 2/3.
 
Ladies and gentlemen,

This argument pops up everywhere on the internet, and has done so for many years. It mostly comes from people never having watched the show. Monty always, always, always, always, always opened up a klunker door on the "reveal" part of the game. It was quite deliberate. It was with knowledge. It is the only way there is any game there at all. Why are you nattering on about assumed intentions, etc., when it is quite clear there would be no tease, and no game if he didn't reveal a klunker every time?

The contestant was always asked to pick one of three doors. Monty always revealed one of the other doors. It always had a klunker behind it. "A year's supply of disposable nosehair trimmers!" He could only have done this having knowledge of where the real prize is. When he removes the known door, the probability space collapses onto the remaining two doors, leaving the contestant's original choice with a 2/3 chance of being wrong. The real fun was watching so many contestants familiar with the game still not get that it was in their best interest to switch to the other door.
 
BillHoyt said:
It mostly comes from people never having watched the show. Monty always, always, always, always, always opened up a klunker door on the "reveal" part of the game. It was quite deliberate.
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The Wikipedia entries on The Monty Hall Problem, Let's Make a Deal and Monty Hall contradict you:
Because of his work on Let's Make a Deal, Hall's name is used in a popular probability puzzle known as the Monty Hall problem. He himself gave a pretty good 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 New York Times reporter John Tierney in 1991
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In the Big Deal, the two contestants were allowed to make a simple choice between three curtains. The top winner in the Big Deal had first choice. One curtain hid the day's Big Deal, which was often multiple cars, a large cash prize, or multiple trips, and typically valued around $10,000.
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As stated, the problem is an extrapolation from the game show: contestants on Let's Make a Deal were not allowed to switch. As Monty Hall wrote to Selvin [1],

And if you ever get on my show, the rules hold fast for you -- no trading boxes after the selection.
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Sorry I didn't get around to posting my spreadsheet, but I realized last night that there was an error in it (though I am not sure exactly what yet, I know there is one) and didn't have time to fix it. And reading some of the new posts since my last led me to this line of thinking, which is the important point I think I was missing: it's not so much which door you pick first, it's that Monty's pick of door is constrained if you pick wrongly to begin with (which you likely will). This makes the second choice not entirely independent of the first.
 
I think JREF posters should be fully aware of exactly what "Wikipedia" is. This disclaimer surely should make its unreliability abundantly clear:

"Wikipedia is an online open-content encyclopedia, that is, a voluntary association of individuals and groups who are developing a common resource of human knowledge. Its structure allows anyone with an Internet connection and World Wide Web browser to alter the content found here.Therefore, please be advised that nothing found here has necessarily been reviewed by professionals with the expertise necessary to provide you with complete, accurate or reliable information.

That's not to say that you won't find valuable and accurate information at Wikipedia, however please be advised that Wikipedia cannot guarantee, in any way whatsoever, the validity of the information found here.It may recently have been changed, vandalized or altered by someone whose opinion does not correspond with the state of knowledge in the particular area you are interested in learning about. "
 
Bill, I've read over the three entries again, and have failed to spot where they contradict each other. I've never seen the show, which might explain my difficulties, so perhaps you (or someone else) can help me.

I'm also well aware of the potential unreliability of wikipedia. When I first read about the Monty Hall problem one of the things that struck me was a mention that it couldn't be applied to the real Let's Make a Deal, so your claim that it could be (and indeed was part of the enjoyment of the show) rather struck me.

I didn't mean to suggest you were wrong (you are of course, an equally (un)reliable internet source), only that there was disagreement on that point - apologies for inadvertantly implying otherwise. If you have any evidence that the Monty Hall problem set-up really does apply to Let's Make a Deal, I would like to see it.
 
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