For a random quantum state on obtained by partial tracing a random pure state on , we consider the question whether it is typically separable or typically entangled. For this problem, we show the existence of a sharp threshold of order roughly . More precisely, for any and for d large enough, such a random state is entangled with very large probability when , and separable with very large probability when . One consequence of this result is as follows: for a system of N identical particles in a random pure state, there is a threshold such that two subsystems of k particles each typically share entanglement if k > k0, and typically do not share entanglement if k < k0. Our methods also work for multipartite systems and for “unbalanced” systems such as , . The arguments rely on random matrices, classical convexity, high-dimensional probability, and geometry of Banach spaces; some of the auxiliary results may be of reference value. © 2013 Wiley Periodicals, Inc.