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Question
If A and B are two events such that \[ P\left( A \right) = \frac{7}{13}, P\left( B \right) = \frac{9}{13} \text{ and } P\left( A \cap B \right) = \frac{4}{13}, \text{ then find } P\left( \overline{ A }|B \right) . \]
Solution
We have ,
\[P\left( A \right) = \frac{7}{13}, P\left( B \right) = \frac{9}{13} \text{ and } P\left( A \cap B \right) = \frac{4}{13}\]
\[As, P\left( \overline{ A } \cap B \right) = P\left( B \right) - P\left( A \cap B \right)\]
\[ \Rightarrow P\left( \overline{ A }\cap B \right) = \frac{9}{13} - \frac{4}{13}\]
\[ \Rightarrow P\left( \overline{ A } \cap B \right) = \frac{9 - 4}{13}\]
\[ \Rightarrow P\left(\overline{ A } \cap B \right) = \frac{5}{13}\]
\[\text{ Now } , \]
\[P\left( \overline{ A }|B \right) = \frac{P\left( \overline{ A } \cap B \right)}{P\left( B \right)} = \frac{\left( \frac{5}{13} \right)}{\left( \frac{9}{13} \right)} = \frac{5}{9}\]
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