Proof:Bayes' rule

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Revision as of 08:43, 3 May 2018 by {{GENDER:Mdg39|[math]'"2}} [/math]'"7
(<math>1) </math>2 | <math>3 (</math>4) | <math>5 (</math>6)
[math]\href{/cs2800/wiki/index.php/Pr}{Pr}(A \href{/cs2800/wiki/index.php/%5Cmid}{\mid} B) = \frac{\href{/cs2800/wiki/index.php/Pr}{Pr}(B \href{/cs2800/wiki/index.php/%5Cmid}{\mid} A) \href{/cs2800/wiki/index.php/Pr}{Pr}(A)}{\href{/cs2800/wiki/index.php/Pr}{Pr}(B)} [/math]
Proof:

By definition, [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(A \href{/cs2800/wiki/index.php/%5Cmid}{\mid} B) = \href{/cs2800/wiki/index.php/Pr}{Pr}(A \href{/cs2800/wiki/index.php/%E2%88%A9}{∩} B) / \href{/cs2800/wiki/index.php/Pr}{Pr}(B) [/math] and [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(B \href{/cs2800/wiki/index.php/%5Cmid}{\mid} A) = \href{/cs2800/wiki/index.php/Pr}{Pr}(A \href{/cs2800/wiki/index.php/%E2%88%A9}{∩} B) / \href{/cs2800/wiki/index.php/Pr}{Pr}(A) [/math]. Multiplying by the denominators gives [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(A \href{/cs2800/wiki/index.php/%5Cmid}{\mid} B) \href{/cs2800/wiki/index.php/Pr}{Pr}(B) = \href{/cs2800/wiki/index.php/Pr}{Pr}(A \href{/cs2800/wiki/index.php/%E2%88%A9}{∩} B) = \href{/cs2800/wiki/index.php/Pr}{Pr}(B \href{/cs2800/wiki/index.php/%5Cmid}{\mid} A)\href{/cs2800/wiki/index.php/Pr}{Pr}(A) [/math]. Dividing by [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(B) [/math] gives the result.

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