A fascinating mathematical problem
Posted by: Minky on 18 April 2004
I've just finished reading "The curious incident of the dog in the night-time" by Mark Haddon. The book has a brief passage on something called "The Monty Hall" problem. Marilyn vos Savant, the smartest person in the world by virtue of her Guinness record IQ, was asked the following question (my words) :
In a game show there are three doors. Behind two of them are goats. Behind the third is a car. You are a contestant trying to win the car. You choose a door and the game show host opens one of the two remaining doors to reveal a goat. Which door is most likely to have the car behind it (should you stick or switch) ?
If you have never encountered this problem before, have a think about it.
When Marilyn vos Savant gave her answer she was quite soundly chastised by some very heavy-hitting brainiacs, and this is at least as fascinating as the problem itself, because (as I have also found) this baby seems to have an almost supernatural ability to turn otherwise mild-mannered people into red-eyed monsters
In a game show there are three doors. Behind two of them are goats. Behind the third is a car. You are a contestant trying to win the car. You choose a door and the game show host opens one of the two remaining doors to reveal a goat. Which door is most likely to have the car behind it (should you stick or switch) ?
If you have never encountered this problem before, have a think about it.
When Marilyn vos Savant gave her answer she was quite soundly chastised by some very heavy-hitting brainiacs, and this is at least as fascinating as the problem itself, because (as I have also found) this baby seems to have an almost supernatural ability to turn otherwise mild-mannered people into red-eyed monsters
I actually came to the forum because my poor little brain wasn't working well enough to attempt to work on my OU maths courses. So why on Earth did I come to this thread? 
I've tried to see how the assumption that the host deliberately chooses a door with a goat might make a difference but I can't, I'm afraid. Maybe it will all make sense in the morning...
Oh! Wait a minute... The goats are indistinguishable in either case but when the host deliberately chooses a door with a goat behind it the doors cease to be indistinguishable (which they were for the problem as originally stated). In this case, we get:
contestant's door..c... g... g
red door (say)......g... c... g
blue door (say).....g... g... c
This time, unlike in my answer to the question as stated, the middle column is also possible because the host chooses the blue door instead of the red door.
So in this case the contestant should switch doors.
[Or I am even more confused than I think]
[This message was edited by JeremyD on Mon 19 April 2004 at 22:12.]
I've tried to see how the assumption that the host deliberately chooses a door with a goat might make a difference but I can't, I'm afraid. Maybe it will all make sense in the morning...
Oh! Wait a minute... The goats are indistinguishable in either case but when the host deliberately chooses a door with a goat behind it the doors cease to be indistinguishable (which they were for the problem as originally stated). In this case, we get:
contestant's door..c... g... g
red door (say)......g... c... g
blue door (say).....g... g... c
This time, unlike in my answer to the question as stated, the middle column is also possible because the host chooses the blue door instead of the red door.
So in this case the contestant should switch doors.
[Or I am even more confused than I think]
[This message was edited by JeremyD on Mon 19 April 2004 at 22:12.]
Hehe Don, I just reread the Brain Teaser thread and saw I was very hard to convince that it's 1/3 and 2/3, I was also sure it's 50%...
I don't have a copy of that spreadsheet, I can only testify it was neat
Omer.
I don't have a copy of that spreadsheet, I can only testify it was neat
Omer.
quote:Since this post I've revised my previous post, now claiming that the contestant should switch if the host deliberately chooses a door with a goat. However, the question does not appear to say this so I still think "it makes no difference" is the answer to the question as stated.
Originally posted by Don Atkinson:
JerremyD, Markus, jayd, Rasher,
does the 1,000,000 door version help?
Cheers
Don
NB If I wake up tomorrow morning and realise that what I've written is rubbish, I reserve the right to claim I was simply stirring.
Matthew, Don, and other code warriors:
How does the code change if the problem is goat-badger-car? Just curious. I've no doubt that your simulations accurately model the problem as you see it. I still maintain that the flaw is in the definition.
Can anybody explain to me how lines 1 and 3 in John Channing's explanation differ (see p.2 of this thread)? I still maintain that, if you can treat the goat separately in lines 1 and 3, there must necessarily be two versions of line 2 (which puts win/lose at 50/50). And if you can't, well then... you're still at 50/50.
Either goat = goat always, or never. Half the time is cheating.
How does the code change if the problem is goat-badger-car? Just curious. I've no doubt that your simulations accurately model the problem as you see it. I still maintain that the flaw is in the definition.
Can anybody explain to me how lines 1 and 3 in John Channing's explanation differ (see p.2 of this thread)? I still maintain that, if you can treat the goat separately in lines 1 and 3, there must necessarily be two versions of line 2 (which puts win/lose at 50/50). And if you can't, well then... you're still at 50/50.
Either goat = goat always, or never. Half the time is cheating.
quote:
Doesn't matter whether you use a goat and a badger or two goats.
Well, there's something we agree on:
Pick | Reveal | Action | Outcome
goat | badger | switch | switch wins
goat | badger | stick | stick loses
badger | goat | switch | switch wins
badger | goat | stick | stick loses
car | badger | switch | switch loses
car | badger | stick | stick wins
car | goat | switch | switch loses
car | goat | stick | stick wins
Did I miss any outcomes? (Bearing in mind that we've already ruled out revealing the car as an intermediate step, thus eliminating half the "pick goat first" and half the "pick badger first" outcomes).
Simple, indeed.
quote:
Originally posted by Don Atkinson:
_Can anybody explain to me how lines 1 and 3 in John Channing's explanation differ _
IF you lable the doors A,B,C(simply for refernce, and the fact that they are there)
In line 1 the car is behind Door C
In line 3 the car is behind Door B
Does this help?
Cheers
Don
Yes. I was afraid you would say that.
Position isn't a factor; the order in which the items occupy the stage has no bearing on the possible outcomes. Ergo, lines 1 and 3 are identical. And we're back to even odds.
Please, let it stop!!!
If you want to try and argue that they are the same then you might as well say that CGG = GCG = GGC because they all have 2 goats and therefore they all cancel and you either have 0% or 100% chance of picking the car.
quote:
Originally posted by jayd:
Position isn't a factor; the order in which the items occupy the stage has no bearing on the possible outcomes. Ergo, lines 1 and 3 are identical. And we're back to even odds.
If you want to try and argue that they are the same then you might as well say that CGG = GCG = GGC because they all have 2 goats and therefore they all cancel and you either have 0% or 100% chance of picking the car.
quote:
Originally posted by John Sheridan:
Please, let it stop!!!quote:
Originally posted by jayd:
Position isn't a factor; the order in which the items occupy the stage has no bearing on the possible outcomes. Ergo, lines 1 and 3 are identical. And we're back to even odds.
If you want to try and argue that they are the same then you might as well say that CGG = GCG = GGC because they all have 2 goats and therefore they all cancel and you either have 0% or 100% chance of picking the car.
Ha. Whole lotta things wrong with that... but you were only joking.
(right?)
Still waiting to hear which outcomes I've omitted from my scenario. What's missing that makes switching such an obvious choice?
Bueller? Bueller? Anyone?