17. (5b, 10-11) In the center of a rectangular billiard table that is 3 meters long and 1 meter wide, a billiard ball is placed. It is struck with a cue in a random direction. After the strike, the ball stops after traveling exactly 2 meters. Find the expected number of reflections from the table edges.
# Solution.

Let's draw a grid with a cell of 3 m $\times 1$ m. The central cell of the grid is the table, highlighted in light green. We need to extend the grid in all directions so that it can accommodate a circle with a radius of 2 m centered at the center of the table $O$.

Allow the ball to "pass through the edges," rolling straight along the grid. Reflection off the edge of the table corresponds to crossing one grid line. Clearly, we can consider only one quarter of the circle, i.e., all directions leading the ball to one of the points on the arc $A K$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0cd9b72133447262cb4bg-13.jpg?height=1265&width=1511&top_left_y=124&top_left_x=318)

From the diagram, it is clear that there are 2 intersections if the ball hits the arc $A B$ or the arc $C L$. If the ball hits the arc $B C$ or $K L$, there is 1 intersection.

$$
\begin{gathered}
\overparen{A B}=\arccos \frac{O M}{O B}=\arccos \frac{3}{4} \\
\overparen{C L}=\arccos \frac{O P}{O L}-\arcsin \frac{O N}{O C}=\arccos \frac{1}{4}-\arcsin \frac{3}{4}
\end{gathered}
$$

Thus, the probability of two intersections is

$$
p_{2}=\frac{2}{\pi}\left(\arccos \frac{3}{4}+\arccos \frac{1}{4}-\arcsin \frac{3}{4}\right) .
$$

Similarly, $\overparen{C B}=\arcsin \frac{3}{4}-\arccos \frac{3}{4}, \quad \overparen{K L}=\arccos \frac{1}{4} . \quad$ Therefore, the probability of one intersection is

$$
p_{2}=\frac{2}{\pi}\left(\arccos \frac{1}{4}+\arcsin \frac{3}{4}-\arccos \frac{3}{4}\right) \text {. }
$$

The expected number of intersections:

$$
p_{1}+2 p_{2}=\frac{2}{\pi}\left(3 \arccos \frac{1}{4}-\arcsin \frac{3}{4}+\arccos \frac{3}{4}\right) .
$$

Answer: $\frac{2}{\pi}\left(3 \arccos \frac{1}{4}-\arcsin \frac{3}{4}+\arccos \frac{3}{4}\right)$.