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Description
Igły Buffona - Buffon's Needles
Problem Buffona o igłach: jeśli mamy igłę o długości L i upuszczamy ją na papier w linie o odstepach D, to jakie jest prawdopodobieństwo, że igła przetnie linie?
Buffon's Needle problem:
Let us say that we have a needle of the length L and we drop it on a paper sheet with lines, where D is the distance between them. what are the chances that the needle will cross any line?
Problem Buffona o igłach: jeśli mamy igłę o długości L i upuszczamy ją na papier w linie o odstepach D, to jakie jest prawdopodobieństwo, że igła przetnie linie?
Buffon's Needle problem:
Let us say that we have a needle of the length L and we drop it on a paper sheet with lines, where D is the distance between them. what are the chances that the needle will cross any line?
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Comments2
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I believe the answer can be found easily in terms of L, D, and θ where θ is the angle between the needle and the lines, if that angle is known. Of course, θ can't be known ahead of time, so to determine the overall probability, we must use calculus and integrate that result over 0 ≤ θ < 2π. This has already been done by many mathematicians in the past, so I'll copy from Wikipedia, substituting the variables you've specified:
P = (2/π)cos-1(D/L) + (2L/πD){1-√[1-(D/L)2]}, if L > D;
P = (2L/πD), if L ≤ D.
Note, when you do that inverse cosine, you'll have to make sure your calculator is in radians mode.
P = (2/π)cos-1(D/L) + (2L/πD){1-√[1-(D/L)2]}, if L > D;
P = (2L/πD), if L ≤ D.
Note, when you do that inverse cosine, you'll have to make sure your calculator is in radians mode.