Let $E$ be an elliptic curve defined over a finite field ${\bf F}_p,$ where $p$ is prime. From Hasse theorem we get $p+1-2\sqrt{p} \leq |E({\bf F}_p)|\leq p+1+2\sqrt{p}.$ Now say that we choose in random the coefficients of $E$ from the interval $[0,p)\cap {\bf Z}$ in such way $E$ is elliptic.
Then, (i) can we say anything about $Pr(|E({\bf F}_p)|=k)?$
(ii) The number $|E({\bf F}_p)|$ takes uniformly, all the values in the interval $[p+1-2\sqrt{p},p+1+2\sqrt{p}]\cap {\bf Z}\ ?$
This is remarkably similar to, but not quite, a classical result of Deuring which classifies the number of elliptic curves over a finite field with a given number of points up to isomorphism. I suggest you read this paper of Schoof. This is not enough to give the full probability, but it does do a lot of the work. It also shows that part (ii) of your question has a negative answer.
[1] Schoof, R, Nonsingular planar cubics over finite fields, J. Combin. Theory Series A 46. (1987), no. 2, 183-211.