Deﬁne a number to be $boring$ if all the digits of the number are the same. How many positive integers less than $10000$ are both prime and boring?
To determine how many positive integers less than $10000$ are both prime and boring, we need to analyze the properties of boring numbers and check their primality.

1. **Definition of Boring Numbers**:
   A number is defined as boring if all its digits are the same. For example, $1, 22, 333, 4444$ are boring numbers.

2. **Range of Boring Numbers**:
   We need to consider boring numbers with 1, 2, 3, and 4 digits:
   - 1-digit: $1, 2, 3, 4, 5, 6, 7, 8, 9$
   - 2-digit: $11, 22, 33, 44, 55, 66, 77, 88, 99$
   - 3-digit: $111, 222, 333, 444, 555, 666, 777, 888, 999$
   - 4-digit: $1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999$

3. **Checking Primality**:
   - **1-digit numbers**: The prime numbers are $2, 3, 5, 7$.
   - **2-digit numbers**: 
     - $11$ is prime.
     - All other 2-digit boring numbers ($22, 33, 44, 55, 66, 77, 88, 99$) are not prime because they are divisible by $11$ or other factors.
   - **3-digit numbers**: 
     - All 3-digit boring numbers ($111, 222, 333, 444, 555, 666, 777, 888, 999$) are divisible by $111$ or other factors, making them non-prime.
   - **4-digit numbers**: 
     - All 4-digit boring numbers ($1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999$) are divisible by $1111$ or other factors, making them non-prime.

4. **Conclusion**:
   - From the above analysis, the only boring number that is prime and less than $10000$ is $11$.

Thus, there is only $\boxed{1}$ positive integer less than $10000$ that is both prime and boring.