Define conditional probability and prove Bayes' rule from it.

easy

Answer

$P(A\; \mid \;B)\; = \;P(A\;\;B)\; / \;P(B)$
$P(B\; \mid \;A)\; = \;P(A\;\;B)\; / \;P(A)$
$P(A\; \mid \;B)\; = \;P(B\; \mid \;A)\; \cdot \;P(A)\; / \;P(B)$
$This is Bayes' rule — a direct consequence of the conditional-probability definition, not a separate axiom.$

Check yourself — multiple choice

$P(A \mid B)\; = \;P(A)$
$P(A \mid B)\; = \;P(A\;\;B) / P(B)$ ; Bayes follows by symmetry: $P(A \mid B)\; = \;P(B \mid A)$ P(A)/P(B)
Conditional prob doesn't exist
$P(A \mid B)\; = \;P(B) / P(A)$

Bayes = direct consequence of the definition of conditional probability.

#probability#bayesian

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