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Bayes' Theorem

mathematics probability statistics
Table of Contents

Bayes’ Theorem is given by the formula:

$$ P(A|B) = \frac{P(B|A)P(A)}{P(B)} $$

Where

  • \(\small P(A|B)\) is the probability of event \(A\) given event \(B\),
  • \(\small P(B|A)\) is the probability of event \(B\) given event \(A\),
  • \(\small P(A)\) is the probability of event \(A\), and
  • \(\small P(B)\) is the probability of event \(B\).

Expanded form #

The denominator can also be expanded using the law of total probability to give:

$$ P(A|B) = \frac{P(B|A)P(A)}{P(B|A)P(A) + P(B|\neg A)P(\neg A)} $$

Where \(\small \neg A\) is the complement of \(\small A\).

Intuition #

This video by 3Blue1Brown was the single biggest help in my understanding of Bayes’ Theorem, you should definitely watch it if you’re struggling to understand it:

Sources #