 In this formula , E is the expected value , pi are the probabilities of attaining each value , and ai are the attainable values . Bernoulli normalizes the expected value by assuming that pi are the probabilities of all the disjoint outcomes of the value , hence implying that p0 + p1 + ... + pn = 1 . Another key theory developed in this part is the probability of achieving at least a certain number of successes from a number of binary events , today named Bernoulli trials , given that the probability of success in each event was the same . Bernoulli shows through mathematical induction that given a the number of favorable outcomes in each event , b the number of total outcomes in each event , d the desired number of successful outcomes , and e the number of events , the probability of at least d successes is 
