> 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?
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).
> 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?
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.
On Nov 2, 9:23 pm, Dan Cass <dc...@sjfc.edu> wrote:
> 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.
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: >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?
> 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
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.
> 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?
> > 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?
> Ignacio Larrosa Caņestro > A Coruņa (Espaņa) > ilarrosaQUITARMAYUSCU...@mundo-r.com
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.
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