If I throw 1000 10-centimeter-long needles on to a tiled floor, where each tile is a 10 cm x 30 cm rectangle, approximately how many needles will end up lying across a crack?
You may assume that the widths of both the needles and the cracks are negligible.
(In reply to re(3): Simulation supports
My solution (neither the first post nor the second thoughts) explicitly mentioned the length of the needle, but, in one instance was taken to be equal to the spacing of the grid lines and the other being 1/3 the spacing of the gridlines.
When the needle is perpendicular to the gridlines that are separated by the same length as the needle, the probability is 1 that it will intersect a line (i.e., the probability is zero so as to be exactly placed to come between the lines). When the needle is parallel to the grid lines, the probability is zero of crossing one. And in intermediate orientations, the presented band -- the width perpendicular to the gridlines -- is sin(x) as a fraction of the width.
For the width of gridlines that is three times the length of the needle, all these projections are 1/3 what they had been, and the correlation of the one to the other is sin(x) vs cos(x).
Posted by Charlie
on 2005-01-21 19:45:40