Example 9. Two steamships must approach the same pier. The arrival time of both steamships is independent and uniformly distributed over the given day. Determine the probability that one of the steamships will have to wait for the pier to be freed, if the first steamship's stay time is one hour, and the second's is two hours.
Solution. Let $x$ and $y$ be the arrival times of the steamships. Possible values for $x$ and $y: 0 \leq x \leq 24, 0 \leq y \leq 24$. Favorable values are $y-x \leq 1, x-y \leq 2$. These inequalities define the area shaded in Fig. 1.13. The area of this region is $S_{g}=24 \cdot 24-0.5 \cdot 23 \cdot 23-0.5 \cdot 22 \cdot 22=69.5$. Since $S_{G}=24 \cdot 24=576$,

$$
P=\frac{S_{g}}{S_{G}}=\frac{69.5}{576} \approx 0.121
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_720b3cbdc89c57e2b629g-034.jpg?height=769&width=772&top_left_y=379&top_left_x=176)

Fig. 1.13