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Question
Mark the correct alternative in the following question:
\[\text{ Let A and B be two events such that P } \left( A \right) = 0 . 6, P\left( B \right) = 0 . 2, P\left( A|B \right) = 0 . 5 . \text{ Then } P\left( \overline{A}|\overline{B} \right) \text{ equals } \]
Options
\[ \frac{1}{10} \]
\[\frac{3}{10} \]
\[\frac{3}{8} \]
\[\frac{6}{7}\]
Solution
\[\text{ We have } , \]
\[P\left( A \right) = 0 . 6, P\left( B \right) = 0 . 2, P\left( A|B \right) = 0 . 5\]
\[\text{ As} , P\left( A|B \right) = 0 . 5\]
\[ \Rightarrow \frac{P\left( A \cap B \right)}{P\left( B \right)} = 0 . 5\]
\[ \Rightarrow P\left( A \cap B \right) = 0 . 5 \times P\left( B \right)\]
\[ \Rightarrow P\left( A \cap B \right) = 0 . 5 \times 0 . 2\]
\[ \Rightarrow P\left( A \cap B \right) = 0 . 1\]
\[\text{ Also } , P\left(\overline{ B } \right) = 1 - P\left( B \right) = 1 - 0 . 2 = 0 . 8\]
\[\text{ As } , P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cap B \right) = 0 . 6 + 0 . 2 - 0 . 1 = 0 . 7\]
\[\text{ Now} , \]
\[P\left( \overline{A}|\overline{B} \right) = \frac{P\left( \overline{A} \cap\overline{ B } \right)}{P\left(\overline{ B} \right)}\]
\[ = \frac{P\left( \overline{A \cup B} \right)}{0 . 8}\]
\[ = \frac{1 - P\left( A \cup B \right)}{0 . 8}\]
\[ = \frac{1 - 0 . 7}{0 . 8}\]
\[ = \frac{0 . 3}{0 . 8}\]
\[ = \frac{3}{8}\]
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