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In probability theory, Buffon's needle problem is a problem first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon. It is a classic problem in geometric probability that can be used to estimate the value of $\pi$.

Problem Statement

Consider a floor with equally spaced parallel lines a distance $D$ apart. A needle of length $L$ (with $L \le D$) is dropped randomly onto the floor. We aim to calculate the probability $P$ that the needle will cross one of the lines.

$Figure: Consider a floor with equally spaced parallel lines a distance $D$ apart. A needle of length $L$ (with $L \le D$) is dropped randomly onto the floor. We aim to calculate the probability $P$ that the needle will cross one of the lines.$

Figure: Consider a floor with equally spaced parallel lines a distance $D$ apart. A needle of length $L$ (with $L \le D$) is dropped randomly onto the floor. We aim to calculate the probability $P$ that the needle will cross one of the lines.

Assumptions

The distance between the lines is $D$.
The length of the needle is $L$.
The needle is dropped randomly in both position and orientation.

Variables

Let $\theta$ be the angle the needle makes with the horizontal (measured in radians).
Let $x$ be the distance from the center of the needle to the nearest line.

Range of Variables

$\theta \in [0, \pi]$ because the needle can point in any direction from 0 to $\pi$.
$x \in [0, D/2]$ because the center of the needle can be at most $D/2$ away from the nearest line to ensure that the needle can cross that line.