Problem 758

Let the times at which the two people arrive at the meeting place be $x$ and $y$. According to the problem, they must satisfy $|x-y| \leqslant \frac{1}{3}$ to meet.

We use their arrival times as the horizontal and vertical coordinates, respectively. Thus, the times at which the two people arrive are uniformly distributed within a square $I$ with a side length of 1 (as shown in Figure 12-5). The meeting phenomenon occurs within the shaded region $G$, where the arrival times $(x, y)$ of person A and person B satisfy $|x-y| \leqslant \frac{1}{3}$. Therefore, the probability that the two people meet is the ratio of the area of region $G$ to the area of region $I$:
$$P=\frac{S_{G}}{S_{I}}=\frac{1-\left(\frac{2}{3}\right)^{2}}{1}=\frac{5}{9}.$$

Thus, the probability that they meet is $\frac{5}{9}$.