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Question
Choose the correct alternative in the following question: \[\text{ Let } P\left( A \right) = \frac{7}{13}, P\left( B \right) = \frac{9}{13} \text{ and } P\left( A \cap B \right) = \frac{4}{13} . \text{ Then } , P\left( \overline{ A }|B \right) = \]
Options
\[ \frac{5}{9}\]
\[\frac{4}{9}\]
\[\frac{4}{13}\]
\[ \frac{6}{13}\]
Solution
We have ,
\[P\left( A \right) = \frac{7}{13}, P\left( B \right) = \frac{9}{13} \text{ and } P\left( A \cap B \right) = \frac{4}{13}\]
\[\text{ As } , P\left( \overline{ A } \cap B \right) = P\left( B \right) - P\left( A \cap B \right)\]
\[ = \frac{9}{13} - \frac{4}{13}\]
\[ = \frac{5}{13}\]
\[ \text{ Now } , \]
\[P\left( \overline{ A }|B \right) = \frac{P\left( \overline{A }\cap B \right)}{P\left( B \right)}\]
\[ = \frac{\left( \frac{5}{13} \right)}{\left( \frac{9}{13} \right)}\]
\[ = \frac{5}{9}\]
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