How many cards should you lay out in a game of EvenQuads?: A detailed study of caps in AG(n,2)

We define a $\mathit{\text{cap}}$ in the affine geometry $AG(n,2)$ to be a subset in which any collection of 4 points is in general position. In this paper we classify, up to affine equivalence, all caps in $AG(n,2)$ of size $k\le 9$. As a result, we obtain a complete characterization of caps in dimension $n\le 6$, in particular complete and maximal caps. Since the $\mathit{\text{EvenQuads}}$ card deck is a model for $AG(6,2)$, as a consequence we determine the probability that an arbitrary $k$-card layout contains a quad.