16. Figure 2 is a street map composed of $3 \times 4$ unit squares, with lines representing roads. Person A starts from point $A(0,0)$ and walks to point $B(4,3)$ along the shortest path, while person B starts from point $B$ and walks to point $A$ along the shortest path. If they start at the same time and move at the same speed, what is the probability that A and B will meet on the road?
16. Suppose A and B take $t=1$ to travel a unit length of a line segment.

As shown in Figure 6, it is not difficult to calculate that when A starts from point $A$ and $t=3$, the probabilities of reaching points $P_{1}(0,3)$, $P_{3}(1,2)$, $P_{5}(2,1)$, and $P_{7}(3,0)$ are
$$
\frac{1}{8}, \frac{3}{8}, \frac{3}{8}, \frac{1}{8}
$$

Similarly, when B starts from point $B$ and $t=3$, the probabilities of reaching points $P_{2}(1,3)$, $P_{4}(2,2)$, $P_{6}(3,1)$, and $P_{8}(4,0)$ are
$$
\frac{1}{8}, \frac{3}{8}, \frac{3}{8}, \frac{1}{8} \text {. }
$$

The probability that A and B meet on the segment $P_{1} P_{2}$ is:
the probability that A passes through the segment $P_{1} P_{2}$ $\times$ the probability that B passes through the segment $P_{1} P_{2}$, i.e.,
$$
\frac{1}{8} \times\left(\frac{1}{8} \times \frac{1}{2}\right)=\frac{1}{8} \times \frac{1}{16} .
$$

Similarly, the probabilities that A and B meet on the segments $P_{3} P_{2}$, $P_{3} P_{4}$, $P_{5} P_{4}$, $P_{5} P_{6}$, $P_{7} P_{6}$, and $P_{7} P_{8}$ are $\frac{3}{16} \times \frac{1}{16}$, $\frac{3}{16} \times \frac{3}{16}$, $\frac{3}{16} \times \frac{3}{16}$, $\frac{3}{16} \times \frac{3}{16}$, $\frac{1}{16} \times \frac{3}{16}$, and $\frac{1}{16} \times \frac{1}{8}$, respectively.
In summary, the probability that A and B meet on the road is
$$
\begin{array}{l} 
\frac{1}{8} \times \frac{1}{16}+\frac{3}{16} \times \frac{1}{16}+\frac{3}{16} \times \frac{3}{16}+\frac{3}{16} \times \frac{3}{16}+\frac{3}{16} \times \\
\frac{3}{16}+\frac{1}{16} \times \frac{3}{16}+\frac{1}{16} \times \frac{1}{8}=\frac{37}{256} .
\end{array}
$$
(Provided by the Popularization Committee of the Sichuan Mathematical Society)