Buffon's Needle Problem Revisited 4

Probability Level 3

Suppose you drop a needle onto a square grid. The center of the needle is going to land inside a grid square $G$ (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 both the left and bottom edges of this grid square $G$ ? (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 0.0796.

1 solution

Arjen Vreugdenhil
Dec 22, 2017

The center of the needle falls in one of four quarter of the square with equal probability.

If in the bottom-left quarter, it crosses both left and bottom edge with probability $1/\pi$ .
In the other three quarters, it never crosses both left and bottom edge.

Taking the average, the probability is $\frac{1 + 3\times 0}{4\pi} = \frac{1}{4\pi} \approx \boxed{0.0796}.$

Buffon's Needle Problem Revisited 4

The answer is 0.0796.

1 solution

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