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Question
Choose the correct alternative in the following question:
\[\text{ If } P\left( A \right) = \frac{2}{5}, P\left( B \right) = \frac{3}{10} \text{ and } P\left( A \cap B \right) = \frac{1}{5}, \text{ then } , P\left( \overline { A }|\overline{ B } \right) P\left( \overline{ B }|\overline{ A } \right) \text{ is equal to } \]
Options
\[ \frac{5}{6} \]
\[\frac{5}{7}\]
\[ \frac{25}{42}\]
1
Solution
\[\text{ We have } , \]
\[P\left( A \right) = \frac{2}{5}, P\left( B \right) = \frac{3}{10} \text{ and } P\left( A \cap B \right) = \frac{1}{5}\]
\[\text{ Also } , P\left( \overline {A} \right) = 1 - P\left( A \right) = 1 - \frac{2}{5} = \frac{5 - 2}{5} = \frac{3}{5} \text{ and } \]
\[P\left( \overline {B} \right) = 1 - P\left( B \right) = 1 - \frac{3}{10} = \frac{10 - 3}{10} = \frac{7}{10}\]
\[\text{ As } , P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cap B \right)\]
\[ = \frac{2}{5} + \frac{3}{10} - \frac{1}{5}\]
\[ = \frac{4 + 3 - 2}{10}\]
\[ = \frac{5}{10}\]
\[ = \frac{1}{2}\]
\[\text{ Also } , P\left( \overline {A} \cap\overline { B } \right) = P\left( \overline {A \cup B} \right)\]
\[ = 1 - P\left( A \cup B \right)\]
\[ = 1 - \frac{1}{2}\]
\[ = \frac{1}{2}\]
\[\text{ Now } , \]
\[P\left( \overline {A}|\overline {B} \right) \times P\left( \overline {B}|\overline {A} \right) = \frac{P\left( \overline {A} \cap \overline {B} \right)}{P\left( \overline {B} \right)} \times \frac{P\left(\overline {A} \cap \overline {B} \right)}{P\left( \overline {A} \right)}\]
\[ = \frac{\left( \frac{1}{2} \right)}{\left( \frac{7}{10} \right)} \times \frac{\left( \frac{1}{2} \right)}{\left( \frac{3}{5} \right)}\]
\[ = \frac{10}{2 \times 7} \times \frac{5}{3 \times 2}\]
\[ = \frac{5}{7} \times \frac{5}{6}\]
\[ = \frac{25}{42}\]
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