Differences

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--- bayes_theorem [2025/09/21 17:00] – hkimscil
+++ bayes_theorem [2025/09/21 22:35] (current) – hkimscil
@@ Line 6: / Line 6: @@
 \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) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1) \\
+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) & = & P(A \cap B) + P(\neg A \cap B)  \nonumber \\
-& & = P(B \cap A) + P(B \cap \neg A)  \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) \\
+& = & 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*}