====== Bayes' Theorem ======
<WRAP left>
\begin{eqnarray*}
P(A \mid B) & = & \dfrac{P(A \cap B)}{P(B)}  \nonumber \\
P(B \mid A) & = & \dfrac{P(B \cap A)}{P(A)}  \nonumber \\
\text{heance }   \nonumber \\
P(A \cap B) & = & P(A \mid B) * P(B) \;\; \text{ and   }  \nonumber \\
P(B \cap A) & = & P(B \mid A) * P(A) \qquad\qquad\qquad\qquad\qquad\qquad\qquad (1) \\
 \nonumber \\
 \nonumber \\
P(B) & = & P(A \cap B) + P(\neg A \cap B)  \nonumber \\
& = & P(B \cap A) + P(B \cap \neg A)  \nonumber \\
& = & P(B \mid A) * P(A) + P(B \mid \neg A) * P(\neg A) \qquad\qquad (2) \\
\\
\\
P(A \mid B) & = & \dfrac{P(A \cap B)}{P(B)} \nonumber  \\
& = & \dfrac{P(B \cap A)}{P(B)} \;\;\; \text{ from (1) and (2)  } \nonumber  \\
& = & \dfrac {(1)} {(2)} \nonumber \\
& = & \dfrac {P(B \mid A) * P(A)} {P(B \mid A) * P(A) + P(B \mid \neg A) * P(\neg A)}  \qquad\qquad (3) \\


\end{eqnarray*}
</WRAP>
<WRAP clear />



<WRAP box left>
\begin{eqnarray}
& & P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}  \nonumber \\
& & P(B \mid A) = \dfrac{P(B \cap A)}{P(A)}  \nonumber \\
& & \text{heance }   \nonumber \\
& & P(A \cap B) = P(A \mid B) * P(B) \;\; \text{ and   }  \nonumber \\
& & P(B \cap A) = P(B \mid A) * P(A)  \\
 \nonumber \\
 \nonumber \\
& & P(B) = P(A \cap B) + P(\neg A \cap B)  \nonumber \\
& & = P(B \cap A) + P(B \cap \neg A)  \nonumber \\
& & = P(B \mid A) * P(A) + P(B \mid \neg A) * P(\neg A) \\
 \nonumber \\
 \nonumber \\
& & \text{suppose that we not know  }  P(B) \nonumber \\
& & P(A \mid B) = \dfrac{P(A \cap B)}{P(B)} \nonumber  \\
& & = \dfrac{P(B \cap A)}{P(B)} \;\;\; \text{ from (1) and (2)  } \nonumber  \\
& & = \dfrac {(1)} {(2)} \nonumber \\
& & = \dfrac {P(B \mid A) * P(A)} {P(B \mid A) * P(A) + P(B \mid \neg A) * P(\neg A)} \\
\end{eqnarray}
</WRAP>
<WRAP clear />

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