Buffon's Needle Problem Revisited 3
Suppose you drop a needle onto a square grid. The center of the needle is going to land inside a grid square (the center lands on the boundary of a square with zero probability, so we can ignore this). What is the probability that the needle lies across the left or bottom edges of this grid square ? (Find a closed form for this probability.)
Details and Assumptions:
(0) The grid is infinite in the x- and y- directions:
Source: grid image
(1) The length of the needle equals the length of each grid square.
(2) The center of the needle lands anywhere on the grid with equal probability.
(3) The needle can be rotated (around its center) at any angle with equal probability.
(i.e., the position and rotation of the needle are each distributed uniformly, as in Buffon's needle problem)
Inspired by Buffon's needle problem ..
Also see this problem
The answer is 0.557.
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1 solution
The center of the needle falls in one of four quarter of the square with equal probability.
If in the top-right quarter, it will not cross the bottom or left edge.
If in the top-left quarter, it crosses the left edge with probability 2 / π .
If in the bottom-right quarter, it crosses the bottom edge with probability 2 / π .
If in the bottom-left quarter, it crosses left and/or bottom edge with probability 3 / π .
Taking the average, the probability is 4 π 0 + 2 + 2 + 3 = 4 π 7 ≈ 0 . 5 5 7 .