Given n events A 1, A 2, · ··, An , bounds are obtained for the probability of the occurrence of at least m out of the n events. The bounds are linear in terms of S 1 and S 2 where S 1 = Σi P(Ai ) and S2 = Σi<j Ρ (Α i ,Α j ). The bounds are illustrated with an example. For m = 1 the bounds are the same as the best known linear bounds in terms of S 1 and S 2.