Example 3. (Buffon's Problem). A plane is marked with parallel lines, the distance between which is $a$. A segment of length $l(l<a)$ is thrown randomly onto this plane. What is the probability that the segment intersects at least one of the lines in the family?

![](https://cdn.mathpix.com/cropped/2024_05_22_720b3cbdc89c57e2b629g-028.jpg?height=164&width=516&top_left_y=573&top_left_x=24)

Fig. 1.4
Solution. Let the distance from the upper end of the segment to the nearest lower straight line be denoted by $y$ (Fig. 1.4). The angle between the segment and the ray parallel to the family of straight lines, whose origin coincides with the upper end of the segment, is denoted by $x$. Clearly, $0 \leq y \leq a$, $0 \leq x \leq \pi$. For the segment to intersect at least one of the straight lines of the family, it is necessary and sufficient that $y=a$ or $y \leq l \sin x$. The expression "the segment is thrown at random" will be understood as follows: the point $(x, y)$ is thrown at random onto the rectangle: $0 \leq x \leq \pi, 0 \leq y \leq a$ (Fig. 1.5). The points whose coordinates satisfy the inequality $y \leq l \sin x$ form the figure shaded in Fig. 1.5. The area of this figure is

$$
S_{1}=\int_{0}^{\pi} l \sin x d x=-\left.l \cos x\right|_{0} ^{\pi}=2 l
$$

The area of the entire rectangle is $S=a \pi$. By formula (1.5.1), taking $S_{g}=S_{1}, \quad S_{C}=S$, we find the desired probability

$$
P(A)=\frac{S_{1}}{S}=\frac{2 l}{a \pi}
$$

where $A$ is the event "the segment intersects at least one straight line".

Remark. In the case $l=a$, the probability is:

$$
P(A)=\frac{2}{\pi} \approx 0.637
$$