Problem
29 September 2005 - 7:00pm
Imagine you are on some kind of climbing game show (provided they exist). You've done all the climbing requirements and answered all the trivia; now you're at the final round.
You are presented with three doors. Behind one of the doors, you are told (truthfully) by the host, is all the climbing gear you could ever want, and in a brand new 4WD. Behind each of the other two doors is a bogus prize, for example a goat. Obviously, the goal is to correctly choose the door with all the gear behind it. (Those of you from the Waikato may beg to differ.)
So, you choose a door. The host then opens one of the doors that you didn't choose, revealing a goat. He/she then asks if you'd like to change your mind. What do you do? :?
I'll let you guys debate it for a while. If you've already heard this please refrain. It's a beauty I reckon.
(Which will win? Logic or intuition?....) :wink:
29 September 2005 - 7:12pm
#2
doors - what's your final answer, and why? ;)
for a great and funny analysis of this problem, see the book 'a curious incident about a dog in the night-time'.
(a great book with a narrator who is a 15 yr old kid with aspergers syndrome.)
29 September 2005 - 8:13pm
#3
ahhh... I see... do you take the mathematical approach and think of it now as a 50/50 choice? Or do you try and figure out the hosts pysche and wonder why he opened a different door...
29 September 2005 - 10:35pm
#4
*ponders*
Hrm, intresting.
I would say, no I don't change my mind. Now my choice was door X so open it up, 50% chance.
:?:
29 September 2005 - 11:50pm
#5
It depends if I want onre goat or two really
30 September 2005 - 9:27am
#6
what breed of goat are we talking about?
30 September 2005 - 10:51am
#7
Oh yeh
I've heard this one. Wasn't it the 64,000 dollar question?
But I couldn't afford petrol for a brand new 4WD. I'd rather some sort of hippy van that I could live in aswell.
30 September 2005 - 11:04am
#8
Give me the goat
I'll carve off what I want, and ride the rest home
30 September 2005 - 3:38pm
#10
Hmm not really a matter of debate. There really is a right and wrong answer. The explaination is a bit tricky though. *SPOILER* if you want the solution try here http://www.remote.org/frederik/projects/ziege/.
Chris
30 September 2005 - 7:33pm
#12
Thanks chris 8)
1 October 2005 - 10:26am
#13
Chris, how could you? :x
Well I did put a spoiler alert, and it's not that hard to google for :roll: Sorry if I spoilt it for anyone. :oops:
Chris
7 October 2005 - 6:08pm
#14
i'd choose the remaining door that i hadnt already chosen. but then you get into the double bluff etc
8 October 2005 - 10:33am
#16
Gee Rob, I think I´d pay not to have the 4WD........
8 October 2005 - 4:30pm
#17
no, i didnt read all the posts.
12 October 2005 - 10:38pm
#18
hey this is cool...
so i CHANGE DOORS. silly me, what was i thinking. hehe.
why? well i rekon first off you had one third chance getting it right, now you have two thirds chance getting it right. though i doubt if id ever have time to think of that in a game show with a host (those hosts always have such annoying voices)
13 October 2005 - 12:18pm
#19
After reading the "solution" All I can say is bullocks.
There is a fallacy to the mathematics used. Just like you can mathematically prove that 1=0 so long as you don't mind dividing by 0.
13 October 2005 - 12:44pm
#20
but then you get into imaginary numbers and crap, which is where it gets a bit messy (im only in 4th form so i havent spent much time with them, despite the fact that im doing 2 and a bit maths courses this year)
13 October 2005 - 2:26pm
#21
After reading the "solution" All I can say is bullocks.
There is a fallacy to the mathematics used. Just like you can mathematically prove that 1=0 so long as you don't mind dividing by 0.
It's not a very elegant mathematical solution but I can't quite see the fallacy in it. Besides the "intuative" solution seems right to me too.
Chris
13 October 2005 - 2:43pm
#22
There is a fallacy to the mathematics used. Just like you can mathematically prove that 1=0 so long as you don't mind dividing by 0.
That's a pretty strong comparison to make! Dividing anything by zero is 'wrong' in the math world (Jack was on the right track with the imaginary #s, you would have to start making stuff up to get division by 0 to work). What 'fallacy' in the mathematical proof are you referring to :?:
13 October 2005 - 3:08pm
#23
like they say "there's lies, damn lies and statistics..."
13 October 2005 - 4:28pm
#24
It is actually mathematically correct to change your choice however the question may not fully allude to the fact that the show host must know where the prize is and he will always show you one of the doors which you did not pick (and does not contain the prize).
JimFix: After reading the "solution" All I can say is bullocks
Sorry mate, but I think I'll trust the opinion of Marilyn von Savant who held the title of the "highest IQ" in the guinness book of records (she is the one who wrote the "solution").
P.s Re: DanP's suggestion read "the curious incident of the dog in the night-time... tis a corker.
p.p.s Dan, you coming to the Bay this weekend??
13 October 2005 - 5:39pm
#25
well put mr bungle. we'll still trust you on the chemistry though jim. for now :wink:
no bay for me. last weekend in nz, and i'm visiting rellies in the SI and not even climbing. :o andrew nevin called me last night, trying to rustle up more numbers. have fun!
we had a great day at the head on sunday before it rained this week. the long wall is climbable. (triple dyno/swingin dyno/crack route/french perfection/undercling /tower/pillar/tendon terror/OTGDY/bullshit/local rap ) then i fell off split apple, oh dear.
13 October 2005 - 7:46pm
#26
however the question may not fully allude to the fact that the show host must know where the prize is and he will always show you one of the doors which you did not pick (and does not contain the prize).
yes it does
13 October 2005 - 8:37pm
#27
hey this is cool...
so i stick with the same door.
why? well i rekon first off you had one third chance getting it right, now you have two thirds chance getting it right. though i doubt if id ever have time to think of that in a game show with a host (those hosts always have such annoying voices)
so i stick with the same door.
why? well i rekon first off you had one third chance getting it right, now you have two thirds chance getting it right. though i doubt if id ever have time to think of that in a game show with a host (those hosts always have such annoying voices)
This is wrong but with the right reasoning
How'd you manadge that???????
14 October 2005 - 9:16am
#28
lies, damn lies and statistics...
After reading around, it appears that it is in fact correct. My apologies.
There are several simulators that do show this, like This one. However it is the fact that the host knows where the prize is what alters the statistics. Implied but not stated in the original question. I guess this is why it has caught out better statisticians than me :oops:
Here is a running model that shows contrary evidence to the mathematical proof.
14 October 2005 - 10:13am
#29
It's like saying each lotto ball has a 1/40 chance of coming out. Once one lotto ball had dropped, each remaining lotto ball has a 1/40 chance of being second. With only 39 balls this gives a 39/40 of a second ball coming out. In all 500+ draws, they have always had a second ball.
No! You started off good, but I (and I think you) got completely lost by the end there....
I suck at statistics and probability, but I think the lotto ball problem gets a bit more complicated, due to not only the number of balls but also the order in which they are supposed to come out.
Originally each door has a 1/3 chance of being the prize. However, GIVEN that one door is shown to be incorrect the remaining two are equally likely.
To parallel this, imagine that the host opens the door before you chose. Which door do you chose. It doesn't matter as they are both equal.
I think this only applies if the host opens the door before you choose one.
lies, damn lies and statistics...
YES!
Edit: sorry Jim, spent too long typing....just saw your 2nd post.
14 October 2005 - 10:16am
#30
No! You started off good, but I (and I think you) got completely lost by the end there....
Damn, someone did get a chance to reply to that before I corrected it. The "fallacy" is simply that the host alters the statistics by knowing which is the correct door.
14 October 2005 - 10:24am
#31
Damn, someone did get a chance to reply to that before I corrected it. The "fallacy" is simply that the host alters the statistics by knowing which is the correct door.
That is true!
Here is a running model that shows contrary evidence to the mathematical proof.
According to their 'explanation' the 1/3 - 2/3s works if the host knows where the car is, and is not allowed to show it. The mathematical proof is only 'disproven' when the host does not know where the car is. I am a bit wary of that particular running model, because I have no way of checking the source code of it!
14 October 2005 - 10:31am
#32
i saw a running model too, and it always gave the prize if you changed. that's bad stats and defeats the purpose of a careful analysis i think!
anyway, for links to both statistical analysis and models go here:
http://www.math.uiuc.edu/~hildebr/361/cargoat.html
the original statement of the problem is there, and some discussion of its ambiguity in the first link to pdf file.
14 October 2005 - 10:40am
#33
Dividing by zero does not give you imaginary numbers. An imaginary number is one whose square is a real negative number.
The closest thing you can get to from dividing by zero is infinity. But that doesn't work in totality, because if you multiply infinity by zero you can't get back to the specific number you were dividing by zero originally.
15 October 2005 - 4:28pm
#34
youre probably right there, appologies to anyone who cares. i'll show my maths teacher this thread on monday, might get off a lesson :D
Don't think I'll have a problem with this one....
"I had rather wear out than rust out." George Whitfield