Question

Choose the correct alternative in the following question:

\[\text{ If}  P\left( A \right) = \frac{3}{10}, P\left( B \right) = \frac{2}{5} \text{ and } P\left( A \cup B \right) = \frac{3}{5}, \text{ then} P\left( A|B \right) + P\left( B|A \right) \text{ equals } \]

Options

• ` 1/4`

• ` 7/12`

• ` 5/12`

• `1/3`

Answer 1

We have  , 
\[P\left( A \right) = \frac{3}{10}, P\left( B \right) = \frac{2}{5} \text{ and}  P\left( A \cup B \right) = \frac{3}{5}\]
\[As, P\left( A \cup B \right) = \frac{3}{5}\]
\[ \Rightarrow P\left( A \right) + P\left( B \right) - P\left( A \cap B \right) = \frac{3}{5}\]
\[ \Rightarrow \frac{3}{10} + \frac{2}{5} - P\left( A \cap B \right) = \frac{3}{5}\]
\[ \Rightarrow \frac{3 + 4}{10} - P\left( A \cap B \right) = \frac{3}{5}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{7}{10} - \frac{3}{5}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{7 - 6}{10}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{1}{10}\]
\[\text{ Now } , \]
\[P\left( A|B \right) + P\left( B|A \right) = \frac{P\left( A \cap B \right)}{P\left( B \right)} + \frac{P\left( A \cap B \right)}{P\left( A \right)}\]
\[ = \frac{\left( \frac{1}{10} \right)}{\left( \frac{2}{5} \right)} + \frac{\left( \frac{1}{10} \right)}{\left( \frac{3}{10} \right)}\]
\[ = \frac{5}{10 \times 2} + \frac{10}{10 \times 3}\]
\[ = \frac{1}{4} + \frac{1}{3}\]
\[ = \frac{3 + 4}{12}\]
\[ = \frac{7}{12}\]