If the events $A$ and $B$ are independent, if $P\left(A^{\prime}\right)=\frac{2}{3}$ and…
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.
\(\frac{4}{21}\)
\(\frac{1}{21}\)
\(\frac{5}{21}\)
\(\frac{3}{21}\)
Solution
$\begin{aligned}
& P(A)=1-P\left(A^{\prime}\right)=1-\frac{2}{3}=\frac{1}{3} \\
& \text{and } P(B)=1-P\left(B^{\prime}\right)=1-\frac{2}{7}=\frac{5}{7}
\end{aligned}$
Now if \(A\) and \(B\) are independent event then, \(\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B})=\frac{1}{3} \cdot \frac{5}{7}=\frac{5}{21}\)
$