Solvers Statistics

Total Probability Rule Calculator

Instructions: Use this step-by-step Total Probability Rules calculator to compute the probability of an event $A$ , when you know the conditional probabilities of $A$ with respect to a partition of events $B_i$ . Please type in the conditional probabilities of A with respect to the other events, and optionally, indicate the name of the conditioning events in the form below:

More About the Law of Total Probability

The Law of Total Probability is one of the most important theorems in basic Probability theory. It is a result that gives a clear link of how the probability of an event $A$ is composed of these parts based on conditional events that form up the "total" of the probability of the event $A$ .

Now, in mathematical terms, let $\left\{B\right\}_{i=1}^n$ be a partition of the sample space, and let $A$ be an event. Then, the probability of the event A can be partitioned in the following way.

\Pr(A) = \Pr(A | B_1) \Pr(B_1) + \Pr(A | B_2) \Pr(B_2) + ... + \Pr(A | B_n) \Pr(B_n)

The Total Probability Rule is a pivotal theorem in Probability and Statistics, and it is the foundation of other crucial theorems such as the Theorem of Bayes .