Application of Total Probability Formula

In the process of studying practical problems, the application of total probability formula should not only consider the probability P(A) of event A, but also consider the probability of event A in the case of "known event B has occurred". Generally speaking, the probability of the latter is not necessarily the same as that of the former. For the sake of clarity, the probability in the second case is called conditional probability and expressed as P(A|B) or PB(A).

The total probability formula is an important formula in probability theory, which transforms the probability solution of a complex event A into the probability summation of a simple event in different situations. An important content of probability theory is to study how to calculate the probability of complex events from the calculation of some simple events. The total probability formula and Bayesian formula just play such a role. For the more complex event A, if a complete event group B 1, B2 ... can be found, and it is easier to calculate the probability of each B than the conditional probability P(A/Bi), this is because to calculate the probability related to event A, it may be necessary to use the full probability formula and Bayesian formula.