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Let $\alpha(\mathbb{F}_q^d,p)$ denote the maximum size of a general positionset in a $p$-random subset of $\mathbb{F}_q^d$. We determine the order ofmagnitude of $\alpha(\mathbb{F}_q^2,p)$ up to polylogarithmic factors for allpossible values of $p$, improving the previous best upper bounds obtained byRoche-Newton--Warren and Bhowmick--Roche-Newton. For $d \ge 3$ we prove upperbounds for $\alpha(\mathbb{F}_q^d,p)$ that are essentially tight within certainintervals of $p$.  We establish the upper bound $2^{(1+o(1))q}$ for the number of generalposition sets in $\mathbb{F}_q^d$, which matches the trivial lower bound$2^{q}$ asymptotically in exponent. We also refine this counting result byproving an asymptotically tight (in exponent) upper bound for the number ofgeneral position sets with fixed size. The latter result for $d=2$ improves aresult of Roche-Newton--Warren.  Our proofs are grounded in the hypergraph container method, and additionally,for $d=2$ we also leverage the pseudorandomness of the point-line incidencebipartite graph of $\mathbb{F}_{q}^2$.

Random Tur\'an and counting results for general position sets over finite fields