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Author Topic:   Counter-Intuitive Science
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 4 of 182 (599639)
01-09-2011 3:25 PM
Reply to: Message 3 by slevesque
01-09-2011 2:57 PM


a red flame is less energetic then a blue flame.
Definitely, and even more so in the case of stars - I mean, how hot can a blue star honestly look?

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 Message 3 by slevesque, posted 01-09-2011 2:57 PM slevesque has not replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 130 of 182 (600672)
01-16-2011 8:06 AM
Reply to: Message 124 by slevesque
01-16-2011 3:18 AM


Re: Counter-Intuitive Math
Quite funny you mention this since I stumbled upon the it's wiki page this week (apparently it's a well known phenomenon in statistics)
It is an extremely important effect in sampling bias and has resulted in some very dubious statistical results as you can probably imagine by extrapolating on Dr A's example.
Which was the Wiki page?

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cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 132 of 182 (600677)
01-16-2011 9:31 AM
Reply to: Message 125 by slevesque
01-16-2011 3:22 AM


Re: Counter-Intuitive Math
While in the second (seemingly identical) scenario he does not know where the prize is, then it doesn't matter if you change or not, you'll have the same odds of winning.
This is ill-defined - what happens when the presenter picks the car? Does the contestant win? Is the game replayed? Is the contestant unaware of what the presenter randomly picks (trivial case)?

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 Message 125 by slevesque, posted 01-16-2011 3:22 AM slevesque has replied

Replies to this message:
 Message 134 by slevesque, posted 01-16-2011 3:46 PM cavediver has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 135 of 182 (600700)
01-16-2011 4:05 PM
Reply to: Message 134 by slevesque
01-16-2011 3:46 PM


Re: Counter-Intuitive Math
If the presentator, not knowing where the car is, opens the door with the car. The probability your door has the car immediatly goes down to zero. But if, again not knowing, he opens the goat, then your door's probability does go up to 1/2.
I agree. But you still haven't given guidance on how these two results are to be combined.
those who think the initial case gives you a 50/50 chnce won't have any problem coming to the, correct this time, conclusion that it is 50/50 in the seond case
And this is not correct, unless you fully define the situation. How are you dealing with the case where the unknowing presenter opens the door to the car, and how does it affect your answer? You cannot just ignore it.
Why am I so reminded of our exchanges where I claim not to understand what you mean, and you reply that I'm just being argumentative

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 Message 134 by slevesque, posted 01-16-2011 3:46 PM slevesque has replied

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 Message 136 by slevesque, posted 01-16-2011 4:19 PM cavediver has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 137 of 182 (600707)
01-16-2011 4:38 PM
Reply to: Message 136 by slevesque
01-16-2011 4:19 PM


Re: Counter-Intuitive Math
Ok, I'll give more reasoning:
No, your reasoning is fine.
But since I said the presentator did reveal a goat,
Ah, did you? Here's your original statement:
slev writes:
While in the second (seemingly identical) scenario he does not know where the prize is, then it doesn't matter if you change or not, you'll have the same odds of winning.
What I'm stressing here is the mistake in giving a definitive answer to a loosely defined question. I know it sounds anal (does that phrase translate??) but I see it as an endemic problem and an obstical to critical thinking.
If you clearly state the question as: "the presenter doesn't know where the car is, and he opens one of the two doors at random, and he reveals a goat, then should you swap?" then fair enough, the answer is obviously - it doesn't matter.
BUT if you ask: "the presenter doesn't know where the car is, and he opens one of the two doors at random, then should you swap?" then it is no wonder that people are confused!

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 Message 138 by slevesque, posted 01-16-2011 4:54 PM cavediver has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 153 of 182 (601191)
01-19-2011 5:01 AM
Reply to: Message 138 by slevesque
01-16-2011 4:54 PM


Re: Counter-Intuitive Math
Because imagine two almost identical scenario where you have a choice between door A,B,C. You choose A, the presentator opens door C and chose a goat in both scenarios.
Even to this I will object The scenarios are not in anyway identical as in the second situation you are throwing away one possibility. And use of the word "chose" has a different meaning in each situation (in the first, he choses by way of knowledge, in the second, there is no choice - he makes a random guess which provides the choice for him)
I repeat, I'm not arguing with your result or your methodology. I am trying to ensure that the question is set up unambiguously to begin with, because it is pointless saying that people are confused by the situation when the very question introduces most of that confusion!
When you say, ''I know it sounds anal'' and ask if it translates...
Ha - there's me using one colloquialism to ask about another! That will teach me...
I was just asking if "I know it sounds anal" makes sense to you? (given that you have English as a second language) - it means "I know it appears that I am being unreasonably stubborn about this point"

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 Message 138 by slevesque, posted 01-16-2011 4:54 PM slevesque has replied

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cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 160 of 182 (601473)
01-20-2011 6:57 PM
Reply to: Message 158 by slevesque
01-20-2011 5:27 PM


Re: More counterintuitive Maths
Anyone else can solve these problems, see if I'm at least half-right ?
Well, the 2/3 is obvious What is counterintuuitive is that it could be anything else... and that can be a stumbling block, as the obvious answer is 2/3, so that must be the trap, so what must the real answer be?
I'm not sure where you get 0.25 from. I can get 1/3 by considering chords from a fixed point (one of the triangle vertices), and sweeping the chords from the tangent around the circle and back to the tangent. So through the first 60 degrees, the chords grow to the length of the triangle; the second 60 degrees they are longer, and the final 60 degrees they shrink again back to the tangent line (with length 0)
For 1/2, same thing but consider the chords sweeping across the circle, staying parallel to the tangent at one of the vertices. The base of the triangle is 3/4 of the way across the circle, so the chords between 1/4 of the way and 3/4 of the way across must be longer than the base. So 1/2 are longer.
The thing is, you can't use lines to space fill ! Lines are not infinitessimally thin, they have zero width. Hence the "paradox". The whole question is ill-defined anyway. You cannot just "randomly" draw lines across circles without giving some concept of measure, some distribution function. There is no obvious "uniform distribution" that one could just assume is meant by the questioner.

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 Message 158 by slevesque, posted 01-20-2011 5:27 PM slevesque has replied

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cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 172 of 182 (601687)
01-23-2011 4:24 AM
Reply to: Message 171 by slevesque
01-23-2011 2:43 AM


Re: More counterintuitive Maths
Yet I still do feel quite confident about my 1/4 answer
It's actually the method I started to look at when looking for the 1/3 and 1/2 answers - but the methods behind those answers suddenly became obvious and I forgot about this first idea. It is very elegant, especially as it seems to put the chords in one-to-one correspondance with the points of the circle... but there is an issue that occurs to me
And there is still the other issue of distribution function - how are these lines being chosen? Is your answer based on the most natural distribution? For example, your distribution would have 1/10 of all lines drawn being within 1/20 of the radius from the circumference! Does that sound reasonable? (and one half of all lines are within 3/10 of the radius from the circumference)
Edited by cavediver, : No reason given.
Edited by cavediver, : No reason given.

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 Message 174 by slevesque, posted 01-23-2011 3:17 PM cavediver has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 175 of 182 (601733)
01-23-2011 6:06 PM
Reply to: Message 174 by slevesque
01-23-2011 3:17 PM


Re: More counterintuitive Maths
I have been thinking that an infinit number of lines can have there middle-point be at the middle of the circles. While this is not true for any other lines drawn, as their middle points will be unique to them.
That's the issue that was bothering me.
Which seems to be about what you are saying.
Yep
But the question still remains for me: how can we know which one is more valid ?
But none are more valid than any other, becuase the question itself is not sufficiently well-defined.

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