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.

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