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Question
Mark the correct alternative in the following question:
\[\text{ If A and B are two events such that} P\left( A \right) \neq 0 \text{ and } P\left( B \right) \neq 1,\text{ then } P\left( \overline{ A }|\overline{ B }\right) = \]
Options
\[ 1 - P\left( A|B \right)\]
\[ 1 - P\left( \overline{ A }|B \right)\]
\[\frac{1 - P\left( A \cup B \right)}{P\left( B \right)}\]
\[\frac{P\left( A \right)}{P\left( B \right)}\]
Solution
\[\text{ We have } , \]
\[P\left( A \right) \neq 0 \text{ and } P\left( B \right) \neq 1\]
\[\text{ Now } , \]
\[P\left( \overline{A }|\overline{B }\right) = \frac{P\left(\overline{ A } \cap\overline{ B } \right)}{P\left( B \right)}\]
\[ = \frac{P\left( \overline{A \cup B }\right)}{P\left( \overline{B} \right)}\]
\[ = \frac{1 - P\left( A \cup B \right)}{P\left( \overline{ B }\right)}\]
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