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If $A$ and $B$ are two events such that $P(A \cup B)=\frac{3}{4}$, $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{4}, \mathrm{P}(\overline{\mathrm{A}})=\frac{2}{3}$ where $\overline{\mathrm{A}}$ is the complement of
A, then what is $\mathrm{P}(\mathrm{B})$ equal to ?
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A, then what is $\mathrm{P}(\mathrm{B})$ equal to ?
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The correct answer is:
$2 / 3$
Given $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{3}{4}, \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{4}$,
$\mathrm{P}(\overline{\mathrm{A}})=\frac{2}{3} \Rightarrow \mathrm{P}(\mathrm{A})=\frac{1}{3}$
As we know
$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
$\therefore \quad \frac{3}{4}=\frac{1}{3}+\mathrm{P}(\mathrm{B})-\frac{1}{4}$
$\Rightarrow \quad \frac{3}{4}+\frac{1}{4}-\frac{1}{3}=\mathrm{P}(\mathrm{B})$
$\Rightarrow \mathrm{P}(\mathrm{B})=\frac{2}{3}$
$\mathrm{P}(\overline{\mathrm{A}})=\frac{2}{3} \Rightarrow \mathrm{P}(\mathrm{A})=\frac{1}{3}$
As we know
$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
$\therefore \quad \frac{3}{4}=\frac{1}{3}+\mathrm{P}(\mathrm{B})-\frac{1}{4}$
$\Rightarrow \quad \frac{3}{4}+\frac{1}{4}-\frac{1}{3}=\mathrm{P}(\mathrm{B})$
$\Rightarrow \mathrm{P}(\mathrm{B})=\frac{2}{3}$
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