Andrew writes down all of the prime numbers less than $50$. How many times does he write the digit $2$?
1. First, list all the prime numbers less than $50$. These are:
   \[
   2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
   \]

2. Identify the prime numbers that contain the digit $2$:
   - The number $2$ itself contains the digit $2$.
   - The number $23$ contains the digit $2$.
   - The number $29$ contains the digit $2$.

3. Count the occurrences of the digit $2$ in these numbers:
   - The digit $2$ appears once in the number $2$.
   - The digit $2$ appears once in the number $23$.
   - The digit $2$ appears once in the number $29$.

4. Sum the occurrences:
   \[
   1 + 1 + 1 = 3
   \]

Therefore, the digit $2$ appears $3$ times in the list of prime numbers less than $50$.

The final answer is $\boxed{3}$.