Suppose we have some elliptic curve defined over $\mathbb F_p$, with $p$ a large prime. Let $n$ be the number of points on the curve. I am interested in what is currently known about the probability* that $n$ is prime (i.e. some reasonable lower bound on this probability).
$n$ is well-known to be within the "Hasse-Weil interval": $$t := \left| n - (p + 1) \right| \le 2 \sqrt p$$
By PNT there are something like $\frac {4 \sqrt p} {\log p}$ primes in this interval, but obviously the probability distribution of $t$ in this interval is non-uniform, so this does not directly give us a good bound. Is there perhaps some known constant multiple of this that is a correct bound?
*Note: To clarify the meaning of "probability": Suppose that the curve was chosen at random (uniformly) from all the nonsingular elliptic curves of form $y^2 = x^3 + ax + b$ over $\mathbb F_p$.