We consider an allocation of n balls into N cells according to probabilities pi. Assuming that the balls are allocated successively, denote by ϕ(n, N) the number of such balls which go into an already occupied cell. If n = 2 the probability that two balls will occupy the same cell is equal to the so-called match probability MP = p21 +...+ p2N. An upper estimate for the probability P(ϕ(n, N) ≤ m) which depends only on n and M P is derived. Such inequalities are important for estimation of the reliability of DNA fingerprinting, a new method of crime investigation which is currently much debated.
