A two-digit number is chosen at random. What is the probability that it is a perfect square?

There are 90 two-digit numbers, not 99 or 100: using 99 gives $\frac{2}{33}$ and using 100 gives $\frac{3}{50}$. Wrongly including 100 (or 09) as a seventh square gives $\frac{7}{90}$.