Let be the maximum possible size of a point set in general position in a -random subset of . We determine the order of magnitude of up to a polylogarithmic factor by proving the balanced supersaturation conjecture of Balogh and Luo. Our result also resolves a conjecture implicitly posed by Chen, Liu, Nie and Zeng. In the course of our proof, we establish a lemma that demonstrates a “structure vs. randomness” phenomenon for point sets in finite-field linear spaces, which may be of independent interest.