  Assuming a uniform variant of the Hardy-Littlewood prime $k$-tuple
conjecture, we compute moments of the number of primes not exceeding $N$ in
arithmetic progressions to a common large modulus $q$ as $a \pmod{q}$ varies.
As a consequence, depending on the size of $\varphi(q)$ with respect to $N$,
the prime count exhibits a Gaussian or Poissonian law. An interesting byproduct
is that the least prime in arithmetic progressions follows an exponential
distribution, where some unexpected discrepancies are observed for smooth $q$.
