2015
Problem - 1260
Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime?
There are four single digit prime: $2$, $3$, $5$, and $7$. Therefore, totally $4\times 4$ two-digit numbers and $4$ single-digit numbers can be formed using them.
Out of these $16+4=20$ numbers, $8$ are primes: $2$, $3$, $5$, $7$, $23$, $37$, $53$, and $73$. Therefore the answer is $$\frac{8}{20}=\boxed{\frac{2}{5}}$$