Q89 / 180 · KCET 2014 · Mathematics
If the events \( A \) and \( B \) are independent if \( P\left(A^{\prime}\right)=\frac{2}{3} \) and \( P\left(B^{\prime}\right)=\frac{2}{7} \), then \( P(A \cap B) \) is equal to
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- Correct answerA\( \frac{5}{21} \)
- B\( \frac{3}{21} \)
- C\( \frac{4}{21} \)
- D\( \frac{1}{21} \)
Solution
Given that, events \( A \) and \( B \) are independent events,
\[ \begin{array}{l} P(A)=\frac{2}{3} \text { and } P(B)=\frac{2}{7} \\ P(A \cap B)=P(A) \cdot P(B) \\ =[1-P(A)][1-P(B)] \\ =\left(1-\frac{2}{3}\right)\left(1-\frac{2}{7}\right) \\ =\frac{1}{3} \cdot \frac{5}{7}=\frac{5}{21} \end{array} \]
\[ \begin{array}{l} P(A)=\frac{2}{3} \text { and } P(B)=\frac{2}{7} \\ P(A \cap B)=P(A) \cdot P(B) \\ =[1-P(A)][1-P(B)] \\ =\left(1-\frac{2}{3}\right)\left(1-\frac{2}{7}\right) \\ =\frac{1}{3} \cdot \frac{5}{7}=\frac{5}{21} \end{array} \]
Hence, the correct answer is option: 'A'.