We derive new upper and lower bounds for probabilities that r or at least r out of n events occur. These bounds are optimal since they can turn to equalities. We describe a method of constructing of such bounds. It can be applied in case of measurable spaces and measures with sign as well. We also obtain bounds for conditional probabilities of combinations of events given σ-field. Averaging of both sides of inequalities for conditional probabilities can yield better bounds for unconditional probabilities.