Question

If P (A) = \[\frac{6}{11},\]  P (B) = \[\frac{5}{11}\]  and P (A ∪ B) = \[\frac{7}{11},\]  find

(i) P (A ∩ B)

(ii) P (A/B)

(iii) P (B/A)

Answer 1

Given: 

\[P\left( A \right) = \frac{6}{11}\]

\[P\left( B \right) = \frac{5}{11} \]

\[P\left( A \cup B \right) = \frac{7}{11}\]

\[\text { (i) P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cap B \right)\]

\[ \Rightarrow \frac{7}{11} = \frac{6}{11} + \frac{5}{11} - P\left( A \cap B \right)\]

\[ \Rightarrow P\left( A \cap B \right) = \frac{6}{11} + \frac{5}{11} - \frac{7}{11} = \frac{4}{11}\]

\[\text{(ii) } P\left( A/B \right) = \frac{P\left( A \cap B \right)}{P\left( B \right)}\]

\[ = \frac{\frac{4}{11}}{\frac{5}{11}}\]

\[ = \frac{4}{5}\]

\[\text {(iii) } P\left( B/A \right) = \frac{P\left( A \cap B \right)}{P\left( A \right)}\]

\[ = \frac{\frac{4}{11}}{\frac{6}{11}}\]

\[ = \frac{4}{6}\]

\[ = \frac{2}{3}\]