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| AI... |
 Posted: Mon Nov 02, 2009 4:47 am |
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| Dan Cass... |
| Posted: Mon Nov 02, 2009 6:23 am |
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[quote]You may already know about Buffon's Needle Problem
http://mathworld.wolfram.com/BuffonsNeedleProblem.html
I am thinking about the following variation
Find the probability that a needle of length "l" will
land on a
circle, given a floor with equally spaced Concentric
circles at a
distance "d" apart?
Here is the drawing which I have made for my
simulation
http://i233.photobucket.com/albums/ee201/vcpandya/Need
leProblem.jpg
[/quote]
So are the radii as suggested in your picture to be d,2d,3d,etc?
Seems like an interesting variation to me, but one problem I see is that of finiteness of a region.
If one takes the region as a square of sidelength N centered at (0,0) and then lets N->infinity,
the answer might differ from another choice of squares which are say of sidelength N with lower left corner at (0,0). |
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| Sjoerd Job... |
| Posted: Mon Nov 02, 2009 12:29 pm |
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On 2009-11-02, AI <vcpandya at (no spam) gmail.com> wrote:
[quote]You may already know about Buffon's Needle Problem
http://mathworld.wolfram.com/BuffonsNeedleProblem.html
I am thinking about the following variation
Find the probability that a needle of length "l" will land on a
circle, given a floor with equally spaced Concentric circles at a
distance "d" apart?
Here is the drawing which I have made for my simulation
http://i233.photobucket.com/albums/ee201/vcpandya/NeedleProblem.jpg
[/quote]
This is quite an interesting problem.
When considering an infinite plane, my guess is that the probability
would be the same as for the normal problem. After all, when the needle
lands far enough from the origin, the situation is quite similar to the
original problem.
I have run trials with l = d = 1, with the tip landing in [-k,k]^2, for
various options of k. The # of trials per k was 100000.
When k gets larger, the result turns out closer to 2/pi.
Kind regards,
Sjoerd Job |
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| AI... |
| Posted: Mon Nov 02, 2009 6:40 pm |
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On Nov 2, 9:23 pm, Dan Cass <dc... at (no spam) sjfc.edu> wrote:
[quote]So are the radii as suggested in your picture to be d,2d,3d,etc?
Seems like an interesting variation to me, but one problem I see is that of finiteness of a region.
[/quote]
You can take it as d,2d,3d etc but that will be a special case,
however radii in my pictur goes as d, d+x,d+2x . . . etc (if you take
x = d then we can have d, 2d, 3d etc)
Sjoerd Job wrote:
[quote]I have run trials with l = d = 1, with the tip landing in [-k,k]^2, for
various options of k. The # of trials per k was 100000.
[/quote]
Really??? I am amazed! Have you designed that simulation in
Mathematica? If yes can you please share it?
Regards,
AI |
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| Sjoerd Job... |
| Posted: Tue Nov 03, 2009 12:41 am |
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On 2009-11-03, AI <vcpandya at (no spam) gmail.com> wrote:
[quote]Sjoerd Job wrote:
I have run trials with l = d = 1, with the tip landing in [-k,k]^2, for
various options of k. The # of trials per k was 100000.
Really??? I am amazed! Have you designed that simulation in
Mathematica? If yes can you please share it?
Regards,
AI
[/quote]
My simulation was quite simple, and did only count where there was
exactly one crossing. I have designed the simulation in Haskell, as that
is my language of choice.
------------ Buffon.hs -----------
module Buffon where
import Data.List
import System.Random
nl = 1 -- relative needle length
-- fall expects one argument
-- the width of the board.
-- As a result, it tells you if the needle crossed a circle exactly once
-- or not.
fall :: Double -> IO Bool
fall w = do
x <- randomRIO (-w,w)
y <- randomRIO (-w,w)
t <- randomRIO (0,2*pi)
let r1 = floor . norm $ (x,y) -- where is the tip?
let r2 = floor . norm $ (x+nl*cos t, y+nl*sin t) -- and the end?
return $ r1 /= r2
buffon :: Double -> Int -> IO Double
buffon w k = do
-- Do k drops on a board of width w
list <- sequence $ replicate k (fall w)
let (ts,fs) = partition id list
let tc = fromInteger . toInteger . length $ ts
let fc = fromInteger . toInteger . length $ fs
return $ tc/(tc+fc)
-------------------------------------
If requested, I could try and code an equivalent in Mathematica.
Here are some results from running "buffon w k" several times.
buffon 1 (10^5) : 0.58701, 0.58749, 0.58817
buffon 2 (10^5) : 0.61548, 0.61256, 0.61560
buffon 10 (10^5): 0.63256, 0.63167, 0.63637
buffon 50 (10^5): 0.63518, 0.63486, 0.63872
My trials with k=10^6 all failed, so I guess I will have to optimize my
code a bit more.
Kind regards,
Sjoerd Job
Kind regards,
Sjoerd Job |
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| Ignacio Larrosa Caņestro... |
| Posted: Tue Nov 03, 2009 6:23 am |
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| AI... |
| Posted: Tue Nov 03, 2009 9:20 am |
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On Nov 3, 4:23 pm, "Ignacio Larrosa Caņestro"
<ilarrosaQUITARMAYUSCU... at (no spam) mundo-r.com> wrote:
[quote]"AI" <vcpan... at (no spam) gmail.com> escribiu na mensaxe
novas:51756fa0-f322-481f-8f7d-c70ae8ca1... at (no spam) u36g2000prn.googlegroups.com....
You may already know about Buffon's Needle Problem
http://mathworld.wolfram.com/BuffonsNeedleProblem.html
I am thinking about the following variation
Find the probability that a needle of length "l" will land on a
circle, given a floor with equally spaced Concentric circles at a
distance "d" apart?
Here is the drawing which I have made for my simulation
http://i233.photobucket.com/albums/ee201/vcpandya/NeedleProblem.jpg
Here is proposed and solved:
http://faculty.missouristate.edu/l/lesreid/Adv63.html
--
Saludos,
Ignacio Larrosa Caņestro
A Coruņa (Espaņa)
ilarrosaQUITARMAYUSCU... at (no spam) mundo-r.com
[/quote]
Great! Thanks for that reference. I did not know that this was already
proposed & solved by someone!!!
I thought about this when I was diddling around with questions where
Pi appears out of no where & Buffon's needle problem is one of those.
A special thanks to Sjoerd Job for writing Mathematica code for
simulation.
I really appreciate all responses.
Regards,
AI |
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