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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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The correct answer is:
\( \frac{5}{21} \)
Given that, events \( A \) and \( B \) are independent events,
\[
\begin{array}{l}
P(A)=\frac{2}{3} \text { and } P(B)=\frac{2}{7} \\
P(A \cap B)=P(A) \cdot P(B) \\
=[1-P(A)][1-P(B)] \\
=\left(1-\frac{2}{3}\right)\left(1-\frac{2}{7}\right) \\
=\frac{1}{3} \cdot \frac{5}{7}=\frac{5}{21}
\end{array}
\]
\[
\begin{array}{l}
P(A)=\frac{2}{3} \text { and } P(B)=\frac{2}{7} \\
P(A \cap B)=P(A) \cdot P(B) \\
=[1-P(A)][1-P(B)] \\
=\left(1-\frac{2}{3}\right)\left(1-\frac{2}{7}\right) \\
=\frac{1}{3} \cdot \frac{5}{7}=\frac{5}{21}
\end{array}
\]
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