A twin prime pair is a pair of primes $(p,q)$ such that $q = p + 2$. The Twin Prime Conjecture states that there are infinitely many twin prime pairs. What is the arithmetic mean of the two primes in the smallest twin prime pair? (1 is not a prime.)

$\textbf{(A) }4$
1. **Identify the smallest twin prime pair:**
   - A twin prime pair \((p, q)\) is defined such that \(q = p + 2\).
   - We need to find the smallest pair of prime numbers that satisfy this condition.
   - The smallest prime number is 2. However, \(2 + 2 = 4\) is not a prime number.
   - The next prime number is 3. Checking \(3 + 2 = 5\), we see that 5 is a prime number.
   - Therefore, the smallest twin prime pair is \((3, 5)\).

2. **Calculate the arithmetic mean of the twin prime pair:**
   - The arithmetic mean of two numbers \(a\) and \(b\) is given by \(\frac{a + b}{2}\).
   - For the twin prime pair \((3, 5)\), the arithmetic mean is:
     \[
     \frac{3 + 5}{2} = \frac{8}{2} = 4
     \]

3. **Conclusion:**
   - The arithmetic mean of the two primes in the smallest twin prime pair is 4.

The final answer is \(\boxed{4}\)