A Full Description of All Commutative Associative Polynomial Operations on Probabilities

When two events are independent, the probability that both events occur is equal to the product p1 * p2 of the probabilities of each of these events. The probability that at least one of these events will occur is equal to p1 + p2 − p1 * p2. In both cases, we have a commutative associative polynomial operation. A natural question is: how can we describe all possible operations of this type? These operations are described in this paper.