Almost independence and irreducibility in simple finite and algebraic groups

We study intersections of the form g1C1∩g2C2, where Ci are conjugacy classes of arbitrary finite simple groups and gi are group elements. We show that, generically, |g1C1∩g2C2|∼|C1||C2|/|G|, which means that the events g1C1,g2C2 are almost independent in G. We also discuss the dimension and the irreducibility of such intersections in simple algebraic groups, and expose the anomaly of SL2. This work is motivated by recent questions of Hrushovski.