**Instructions:** Use this step-by-step Bayes Rule Calculator to reverse conditional probabilities using Bayes' Theorem. We need an event \(A\), and you need to 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 Bayes' Rule

Bayes Rule is one of the critical theorems in Probability and Statistics, because it links a very interest concept of causality and conditional probability.

In other words, Bayes rule links the idea of reversing the direction of a conditionality with a very simple calculation based on a priori information

Mathematicall, let \(\{B\}_{i=1}^n\) be a partition of the sample space, and let \(A\) be an event. Then, Bayes Theorem indicates that

\[\Pr(B_i | A ) = \displaystyle \frac{\Pr(A | B_i) \Pr(B_i) }{\Pr(A | B_1) \Pr(B_1) + \Pr(A | B_2) \Pr(B_2) + ... + \Pr(A | B_n) \Pr(B_n)}\]Observe that by the Total Probability Rule, the value in the denominator is simply \(\Pr(A\).

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