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TakenItSeriously

A Logical Explaination for the mysterious results of Buffon's Needle

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When I first came accross the mystery of Buffon’s needle, it was presented as a mystery because, apparently, nobody could understand why it would result in the value of pi or what the problem of scattered needles had to do with a circle.
 
You might actually do the experiment and find that the results really did statistically converge to pi as the sample size grew larger or you might find the mathematical solution would indeed result in a probability that is exactly equal to pi. You might notice that there is a cosine of the angle between the needle and the lines on the table involved. You might even be able to construct a circle to describe how the cosine function relates to a circle using trigonometry, but even then you still probably wouldn’t truly have a clear and direct physical understanding of why the problem of randomly scattered needles should be related to circles.
 
Here, I don’t present the mathematical solution which you can look up online from a number of sources. Instead I present a simple and logical model that explains the problem in the proper physical context which in turn will make it clear why circles are related to randomly distributed needles.
 
Once again, Once you understand the solution it will seem simple to you as all logical solutions that are properly explained will seem relatively simple compared to the math. Perhaps it will even seem like it should be obvious once you understand it and you may not understand why you didnt think of it before but unless you could physically explain it in fore-site before hearing this solution, then it clearly wasn’t really as obvious in fore-site as it may seem in hind-site.
 
I present this solution to you not just to show off that I have a gift for solving logic problems, but to provide yet another example that shows why logic really is just as important as math and that logic and math are not the same thing. Neither is logic just an alternative method to mathematics for solving problems that can be used as a substitute for math. It actually performs a completely different function from the math as I’ve said many times before: Logic clarifies our understanding of the problem while math quantifies the numerical results of the properties involved that can then be compared to experimental results. In fact math and logic are actually complementary opposites.
 
Another words we cannot truly understand a problem without a logical model that can explain it and we cannot truly know that our understanding is correct without validating the mathematical results with experimental test.
 
Problem:
 
Figure 1: Buffon’s needle is a probabilistic method that can provide a good estimate of π based on random events.
 
Assume that you have a needle that has a length of l and a surface that has parallel lines on it that are all equally spaced at 2l distance apart. 
 
If you toss a needle in a random manor on that surface such that it can land in any arbitrary position and orientation, then the probability that the needle lands inbetween the lines divided by the number of times that the needle will intersect with a line will be equal to pi (π).
 
Another words your results will approximate 3.14... etc. with a sufficient sample size and the larger your sample size the better your approximation of π should be.
 
The mystery of this method is why does it approximate π which we know is a constant that must be somehow related to a circle when there seems to be no circles involved with this method of randomly scattering needles.
 
Logical Explaination:
There is actually a simple logical explaination for this mystery and to understand it more easily I will provide a probabilistically equivalent scenario to illustrate why.
 
Instead of using needles we can use clear plastic discs that have the needle embedded in the disc such that they perfectly bisect the circumference of the discs. After all it will still represent a random position and orientation just as the needles would. In fact they would probably be more random than the needles themselves since needles are not perfectly symetrical and they may be tossed in such a way that may be biased while the disc surrounding the needle would ensure a more random or unbiased result.
 
Given in this new context, it should now be clear that the source of pi is linked to the circular shape of the disc. 
 
Put another way, think of the position of the disks and the orientation of the needles as independant properties. It is the probability distribution of the needle’s orientation that has an even disc like distribution about their center of gravity.  By taking all the angles accross the entire sample space then the orientations of all the needles would stack up to be a random sample of all angles between 0 and 2π or between 0 and π if the needle is symmetrical. So you can see that on average, the orientations of all the needles should combine to be distributed in the shape of a disk.
 
If by some extreme long shot, they did not create a reasonable disc like distribution, then you probably would not get a reasonable approximation of pi as your result.
 
 
 
 
 
 
Edited by TakenItSeriously

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I didn't think that there was much mystery about Buffon's needle, except the fact that so many so-called 'real world' experiments of the problem ended up just a little too perfectly in line with predictions many, many times throughout the years.  

In my opinion, if you want something that is a bit more mysterious, check out Bertrand's paradox. https://en.wikipedia.org/wiki/Bertrand_paradox_(probability) It is my favorite example of 'probability requires you to be extremely careful in your definitions'

 

Edited by Bignose

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17 minutes ago, Bignose said:

I didn't think that there was much mystery about Buffon's needle, except the fact that so many so-called 'real world' experiments of the problem ended up just a little too perfectly in line with predictions many, many times throughout the years.  

In my opinion, if you want something that is a bit more mysterious, check out Bertrand's paradox. https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

It is not particularly mysterious.  All it says is that if you pick something at random, you must define the probability distribution.

 

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The Pics didn’t get into the final draft so here they are now:

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2 hours ago, Bignose said:

I didn't think that there was much mystery about Buffon's needle, except the fact that so many so-called 'real world' experiments of the problem ended up just a little too perfectly in line with predictions many, many times throughout the years.  

In my opinion, if you want something that is a bit more mysterious, check out Bertrand's paradox. https://en.wikipedia.org/wiki/Bertrand_paradox_(probability) It is my favorite example of 'probability requires you to be extremely careful in your definitions'

 

I’m not sure I would call the Bertrand Paradox all that mysterious so much as confusing.

I wouldn’t accept any of the three premises given as valid. Premises should be axiomatic and self evident as far as being valid and the premises given are definitely not axiomatic.

My guess is that they are a problem with boundary conditions combined with conflicting properties that cannot be both defined in a non biasing way for both properties at once.

It may not be possible to provide a proper definition for random chords to the circle except perhaps in an infinitely large universe which, of course is not practical.

For a decent approximation, I might try picking random positions within a much larger space than that described by the circle and then assigned to them random vectors for a random distribution of lines that may or may not pass through an arbitrarily defined circle, but I would use a space that was much much larger than the given circle such that vectors that happened to intersect the circle from a long distance away would make an insignificant contribution.

I might also position the circles in a random manor perhaps even using multiple circles with triangles in them.

It’s just my first guess at a reasonable approximation of random that would represent an unbiased distribition or a homogenious isotropic environment FWIW. I’m not sure how well it would do in terms of convergence to a unique result.

Edited by TakenItSeriously

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Edit to add:

Regarding the larger space, I might define a large square for populating the random positions of the universe in a cartesian coordinate system and then I would define a large circle within the boundaries of the square and call it a horizon.

Then I would assign random vectors that pass only through those points within the larger circle so that the points in the corners wouldn't create a bias and the horrizon was always equidistant from the circle. This would then preclude the idea of using multiple circles of course.

 

Edited by TakenItSeriously

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