The Rule
\[ \Pr(x \vert y) = \Pr(y \vert x) \frac{\Pr(x)}{\Pr(y)} \]Conditional Probability
\[ \Pr(x \vert y) = \frac{\Pr(x \land y)}{\Pr(y)} \]Proof
\[ \Pr(x \vert y) = \frac{\Pr(x \land y)}{\Pr(y)} \]
\[ \Rightarrow \Pr(x \land y) = \Pr(x \vert y) \Pr(y) \]
\[ \Pr(y \vert x) = \frac{\Pr(x \land y)}{\Pr(x)} \]
\[ \Rightarrow \Pr(x \land y) = \Pr(y \vert x) \Pr(x) \]
\[ \Rightarrow \Pr(x \vert y) \Pr(y) = \Pr(y \vert x) \Pr(x) \]
\[ \Rightarrow \Pr(x \vert y) = \Pr(y \vert x) \frac{\Pr(x)}{\Pr(y)} \]