16. (1982 American High School Mathematics Examination) Figure 12-6 is a partial map of a city. Small rectangles represent houses and buildings, and the spaces between them represent streets. There is a student who walks from point $A$ to point $B$ every morning along the streets shown in the figure, and he can only walk east or south. At each intersection, he has a probability of $\frac{1}{2}$ of choosing to go east or south (each choice is independent of the others). What is the probability that the student will walk through point $C$?
16. As shown in the figure, according to the walking rules specified in the problem, the student must pass through one of the points $C_{i}(i=0,1,2,3)$ and then move east to reach $C$. That is, the probability that the student passes through $C$ is equal to the sum of the probabilities of passing through each of the points $C_{i}$ and then moving east.

The paths from $A$ to $C_{i}$ contain two eastward segments and $i$ southward segments, thus totaling $2+i$ segments, with the $i$ southward segments appearing in any order.

Therefore, the number of paths from $A$ to $C_{i}$ is $C_{2+i}^{i}$ (or, considering that the two eastward segments can appear in any order, the combination number is $C_{2+i}^{2}$, but $C_{2+i} = C_{2+i}^{2}$).

Since at each intersection, the probabilities of moving east and south are both $\frac{1}{2}$, and the number of intersections (including $A$ and $C_{i}$) on any path from $A$ to $C_{i}$ is $3+i$, the probability of taking this path and moving east is $\left(\frac{1}{2}\right)^{3+i}$. Therefore, the probability sought in the problem is
$$
\begin{aligned}
& \sum_{i=0}^{3} C_{2+i}\left(\frac{1}{2}\right)^{3+i} \\
= & \frac{1}{8}+\frac{3}{16}+\frac{6}{32}+\frac{10}{64} \\
= & \frac{21}{32} .
\end{aligned}
$$

Note: If we write out the probability of passing through each intersection step by step, we can obtain the tree diagram on the upper right (where the final 1 is only for checking the calculation).

From the diagram, we can clearly see that the 20 paths from $A$ to $B$ passing through $C$ (out of a total of 35 paths from $A$ to $B$) are not equally likely.