# Bayes Rule Calculator

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Instructions:
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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

### How do you calculate Bayes' Rule?

Mathematically, 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 \(\Pr(A\).

### How to use this Bayes' Theorem calculator?

This calculator will compute Bayes' rule for you showing all the steps. Essentially what you need to do is to provide the probability of the events that define the partition, and to provide the conditional probability of the event you want to use Bayes for, with respect to the events in the partition.t

### Can I use Bayes Theorem with a tree diagram?

Some people find it clarifying to represent the partition and the corresponding conditional probabilities in the form of a tree diagram. That is certainly of aid to understand things in a more clear way, but it is not really necessary.