Let $P$ be a set of $n$ random points chosen uniformly in the unit square. In this paper, we examine the typical resolution of the order type of $P$. First, we show that with high probability, $P$ can be rounded to the grid of step $\frac{1}{n^{3+\epsilon}}$ without changing its order type. Second, we study algorithms for determining the order type of a point set in terms of the the number of coordinate bits they require to know. We give an algorithm that requires on average $4n \log 2 n + O(n)$ bits to determine the order type of $P$, and show that any algorithm requires at least $4n \log 2 n − O(n \log \log n)$ bits. Both results extend to more general models of random point sets.