proof of Bayes’ Theorem

The proof of Bayes’ theorem is no more than an exercise in substituting the definition of conditional probability into the formula, and applying the total probability theorem.

$$\frac{\mathbb{P}\left\{B\right\}\mathbb{P}\{E|B\}}{{\sum }_i\mathbb{P}\left\{A_i\right\}\mathbb{P}\{E|A_i\}}=\frac{\mathbb{P}\left\{E\cap B\right\}}{{\sum }_i\mathbb{P}\left\{E\cap A_i\right\}}==\frac{\mathbb{P}\left\{E\cap B\right\}}{\mathbb{P}\left\{E\right\}}=\mathbb{P}\{B|E\}.$$

Title	proof of Bayes’ Theorem
Canonical name	ProofOfBayesTheorem
Date of creation	2013-03-22 12:45:09
Last modified on	2013-03-22 12:45:09
Owner	ariels (338)
Last modified by	ariels (338)
Numerical id	5
Author	ariels (338)
Entry type	Proof
Classification	msc 60-00
Classification	msc 62A01