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60-00-ProofOfBayesTheorem.tex
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60-00-ProofOfBayesTheorem.tex
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\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{ProofOfBayesTheorem}
\pmcreated{2013-03-22 12:45:09}
\pmmodified{2013-03-22 12:45:09}
\pmowner{ariels}{338}
\pmmodifier{ariels}{338}
\pmtitle{proof of Bayes' Theorem}
\pmrecord{5}{33058}
\pmprivacy{1}
\pmauthor{ariels}{338}
\pmtype{Proof}
\pmcomment{trigger rebuild}
\pmclassification{msc}{60-00}
\pmclassification{msc}{62A01}
\endmetadata
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%%%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\Prob}[2]{\mathbb{P}_{#1}\left\{#2\right\}}
\newcommand{\norm}[1]{\left\|#1\right\|}
% Some sets
\newcommand{\Nats}{\mathbb{N}}
\newcommand{\Ints}{\mathbb{Z}}
\newcommand{\Reals}{\mathbb{R}}
\newcommand{\Complex}{\mathbb{C}}
\begin{document}
\newcommand{\prob}[1]{\Prob{}{#1}}
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.
$$
\frac{\prob{B}\prob{E|B}}{\sum_i \prob{A_i}\prob{E|A_i}} =
\frac{\prob{E\cap B}}{\sum_i \prob{E\cap A_i}} =
= \frac{\prob{E\cap B}}{\prob{E}} = \prob{B|E}.
$$
%%%%%
%%%%%
\end{document}