# 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:

Probabilities of Partition Events ($$B_i$$'s. Between 0 and 1 and must add up to 1. Comma or space separated) =
Conditional probabilities ($$\Pr(A|B_i)$$'s. Comma or space separated) =
Name of partition events (Optional. Comma separated) =
Name of main event (Optional. Name is $$A$$ by default) =

## 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.

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