# Bayes Rule Calculator

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:

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) =

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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