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Author | Topic: Counter-Intuitive Science | |||||||||||||||||||
Trae Member (Idle past 4327 days) Posts: 442 From: Fremont, CA, USA Joined: |
Saw this on Pharyngula.
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Panda Member (Idle past 3734 days) Posts: 2688 From: UK Joined: |
RAZD writes:
I think the only disagreement we have is in regards to how the question is phrased and not on what the answer is. Curiously, you do not walk away with the car. You still have the option to swap or stay, whether you have a possibility of winning or not.1:3 chance of winning the car. Stay/Swap is 50:50.
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petrophysics1 Inactive Member |
You are correct except that what happened to your 1/3 odds when the goat was seen in one of the doors is your odds for your original choice became 2/3.
If you switch they become 1/2. If you stick they are 2/3, why would you lower your odds? The problem is covered in the movie "21", if you are like me and count cards you will understand it.
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Panda Member (Idle past 3734 days) Posts: 2688 From: UK Joined: |
petrophysics1 writes:
Imagine there were 3 people playing this game: ...except that what happened to your 1/3 odds when the goat was seen in one of the doors is your odds for your original choice became 2/3. You pick Door 1. Monty picks Door 2. I pick Door 3. We all have a 1:3 chance of winning the car.But you are claiming that if Monty randomly shows a goat, then my chances of winning drop and yours increases, which is clearly not true. Remember, this is not the same as the standard Monty Hall question - Monty doesn't know where the car is.
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Noetherian Atheist Junior Member (Idle past 4574 days) Posts: 7 From: London Joined: |
Cavediver,
Well, the 2/3 is obvious Well, I suppose. No, there's no trap. But I find a lot of people expect it to be 1/2. Sorry it was too intuitive!
I'm not sure where you get 0.25 from. Me too. You are "correct" in your approaches to getting 1/2 & 1/3. Clearly this can be extended to any value in between. But 1/4 is new.
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. What? You mean I haven't convinced you that 1/2 = 1/3? Myself, I'd question the whole basis of a probability of choosing any one element of an infinite subset, from a larger set, and finding the answer is "1/3". I thought about an argument based on assuming finitely wide strips, and then limiting, but again, does a probability carry across a limit (or an integral)? Functional analysis is not my area.
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slevesque Member (Idle past 4661 days) Posts: 1456 Joined: |
Myself, I'd question the whole basis of a probability of choosing any one element of an infinite subset, from a larger set, and finding the answer is "1/3". I thought about an argument based on assuming finitely wide strips, and then limiting, but again, does a probability carry across a limit (or an integral)? Functional analysis is not my area. I would guess that it does, or at least that there is a way of finding the correct probability. I remember my physics teacher last year asked us the following math question: given a stick that is a meter long, what is the probability that, by cutting it in 3 parts, I can take those three parts and form triangle. Which is kinda similar to your question, since there are, mathematically, infinitely many ways to cut a stick in three pieces. Yet there was still a real answer. Yet I still do feel quite confident about my 1/4 answer, I have given much thought into it and it still feels like the right approach. While something which I can't put my finger on is missing in the two other reasonings who give 1/2 and 1/3
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cavediver Member (Idle past 3664 days) Posts: 4129 From: UK Joined: |
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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Parasomnium Member Posts: 2224 Joined: |
jar writes: That ice floats. Even more counter-intuitive is that ice also sinks. There are various solid phases of ice, some of which have a density higher than that of liquid water, which causes it to sink. "Ignorance more frequently begets confidence than does knowledge: it is those who know little, not those who know much, who so positively assert that this or that problem will never be solved by science." - Charles Darwin.
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slevesque Member (Idle past 4661 days) Posts: 1456 Joined: |
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 That all points aren't born equal ? 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. (at least that's how I see it, my visualisation could be wrong)
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) I couldn't answer those questions. I actually asked the question to ''Dr.Math'' and they sent me back this answer:
All three answers are equally valid. It depends entirely on how you 'define' a random chord. It is a classic illustration of the fact that 'randomness' is not a completely fixed concept like 'length' or 'weight'. The difficulty in getting a random sample from a population is well-known and accounts for the many occasions that poll ratings fail to predict the actual outcome. The usual definition of a random sample includes the requirement that EVERY member of the population has an equal chance of being in the sample. It is not easy to decide between the three methods of defining a random chord which is the most valid. Which seems to be about what you are saying. I've never had any course in probabilities and statistics, and so when I happen to do problems like these I go with my intuition. This is a case where the reality of the problem was counter-intuitive; randomness isn't just 'randomness', and defining it in different ways will lead to different results all equally valid. But the question still remains for me: how can we know which one is more valid ? If we can extend the survey analogy, there should be only one actual outcome in reality. How can we know which one it will be ?
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cavediver Member (Idle past 3664 days) Posts: 4129 From: UK Joined: |
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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Kaichos Man Member (Idle past 4509 days) Posts: 250 From: Tasmania, Australia Joined: |
One involves 3 plain cards with faces coloured red/red, red/green & green/green. Now if one card is chosen at random and placed on a table so that you can see one side (red say), what it the probability that the other side is also red? 1/3. The chance of it being green is 1/6. The easiest way to comprehend this is to consider that you are always going to bet that the card is the same colour on the other side. Seeing as two of the three cards are the same colour on the other side, your chances of winning are automatically 2/3. I know it sounds strange, but it actually isn't. Another way of seeing it ; 3 cards have six sides in total. Chances of pulling out red/red? 1/3. Multiplied by the chance of it showing red? One. One times 1/3 is 1/3. Chances of pulling out red/green? 1/3. Multiplied by the chance of it showing red? 1/2. 1/2 times 1/3 =1/6. As for the Monty Hall problem, the best way of understanding it is to remember that Monty knows where the car is- so it's kind of like Monty "owning" two of the doors, and you only own one. Monty's always going to have a goat to reveal (there's only one car) but when you switch you are literally exchanging your 1/3 shot for Monty's 2/3. If Monty doesnt know, and reveals a goat, your chances remain at 1/3- change or swap. All it means is that Monty's 1/3 shot revealed a goat. It doesn't change your odds. "When man loses God, he does not believe in nothing. He believes in anything" G.K. Chesterton
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Kaichos Man Member (Idle past 4509 days) Posts: 250 From: Tasmania, Australia Joined: |
or one that can travel directly upwind, powered by wind RAZD, I assume you are referring to a yacht tacking. I hate to be pedantic but that isn't "directly" into the wind. However, it is pretty mindblowing that a wind-powered vessel can travel into the wind, directly or indirectly. My personal favourite is that smoke is heavier than air, and will flow downwards out of a test-tube. "When man loses God, he does not believe in nothing. He believes in anything" G.K. Chesterton
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Kaichos Man Member (Idle past 4509 days) Posts: 250 From: Tasmania, Australia Joined: |
that it is entirely possible to make a vehicle that travels directly downwind, powered only by the wind, and goes faster than the wind Okay Arachnaphilia, you got me. Elaborate. "When man loses God, he does not believe in nothing. He believes in anything" G.K. Chesterton
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arachnophilia Member (Idle past 1365 days) Posts: 9069 From: god's waiting room Joined: |
LMGTFY - Let Me Google That For You
basically, you make a cart such that the wheel powers a propeller (that rotates against the wind). the wind will push the cart because the prop acts as a dead sail at first, but as the wheels power the prop, the the added force of the wind and the prop cause the cart to move faster than the wind. Edited by arachnophilia, : spalling
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RAZD Member (Idle past 1426 days) Posts: 20714 From: the other end of the sidewalk Joined: |
Hi Kaichos Man
RAZD, I assume you are referring to a yacht tacking. I hate to be pedantic but that isn't "directly" into the wind. However, it is pretty mindblowing that a wind-powered vessel can travel into the wind, directly or indirectly. Nope. A vessel with a propeller that is spun by the wind powers a propeller that drives the vessel directly upwind. Same principal as the vehicle going downwind faster than the wind. another idea | Boat Design NetPage Not Found Page Not Found quote: Enjoy. by our ability to understand Rebel American Zen Deist ... to learn ... to think ... to live ... to laugh ... to share. Join the effort to solve medical problems, AIDS/HIV, Cancer and more with Team EvC! (click)
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