Stochastic independence of complex events

The problem of independence of random events is one of the most important and insufficiently studied in probability theory. The importance of the problem is caused by the mass application of the assumption of independence of factors in practical provisions. The article shows formulas for calculating the conditional probabilities of a sum, the product of events and the opposite event. Two necessary and sufficient conditions for the independence of events have been determined. The obtained results can be used in intelligent decision-making systems.