Counting occurences of 'a' in a book faster than O(n)?
You could try sampling. The assumption that the proportion of a's on any given page is roughly the same as on any other one (and that all pages have roughly the same amount of text) seems reasonable. If $n$ is the total number of pages, you could pick, say $k=n^{4/5}$ random pages, count their average number of characters and the proportion of a's, and scale up to $n$. Your runtime would be $O(n^{4/5})$, and with high probability, your estimate for the number of a's would deviate from the true count by $\tilde O(n^{3/5})$.

You can play with the exponents to get other accuracy-runtime tradeoffs (e.g., in $O(n)$ time you can achieve whp a deviation of $\tilde O(\sqrt n)$).

As pointed out in the comments, an exact count requires $\Omega(n)$ runtime. If the interviewer absolutely insisted on a sublinear runtime, I would suggest sampling.