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If $A$ and $B$ are independent events and $P(A)=\frac{2}{3}, P(B)=\frac{3}{5}$, then $P\left(A^{\prime} \cap B\right)=$
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The correct answer is:
$\frac{1}{5}$
$\mathrm{P}(\mathrm{A})=\frac{3}{5}, \mathrm{P}(\mathrm{B})=\frac{2}{3}$ and $\mathrm{A}, \mathrm{B}$ are independent events.
$$
\begin{aligned}
\therefore P(A \cap B) &=P(A) \times P(B)=\frac{3}{5} \times \frac{2}{3}=\frac{2}{5} \\
P(A \cup B) &=P(A)+P(B)-P(A \cap B) \\
&=\frac{3}{5}+\frac{2}{3}-\frac{2}{5}=\frac{13}{15} \\
\therefore P(A \cup B)^{\prime} &=1-\frac{13}{15}=\frac{2}{15} \\
P\left(A^{\prime} \cap B^{\prime}\right) &=P(A \cup B)^{\prime}=\frac{2}{15}
\end{aligned}
$$
$$
\begin{aligned}
\therefore P(A \cap B) &=P(A) \times P(B)=\frac{3}{5} \times \frac{2}{3}=\frac{2}{5} \\
P(A \cup B) &=P(A)+P(B)-P(A \cap B) \\
&=\frac{3}{5}+\frac{2}{3}-\frac{2}{5}=\frac{13}{15} \\
\therefore P(A \cup B)^{\prime} &=1-\frac{13}{15}=\frac{2}{15} \\
P\left(A^{\prime} \cap B^{\prime}\right) &=P(A \cup B)^{\prime}=\frac{2}{15}
\end{aligned}
$$
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