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If $\mathrm{P}(\mathrm{B})=\frac{3}{4}, \mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \overline{\mathrm{C}})=\frac{1}{3}$ and $\mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B} \cap \overline{\mathrm{C}})=\frac{1}{3}$,
then what is $\mathrm{P}(\mathrm{B} \cap \mathrm{C})$ equal to?
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then what is $\mathrm{P}(\mathrm{B} \cap \mathrm{C})$ equal to?
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Verified Answer
The correct answer is:
$\frac{1}{12}$
$\mathrm{P}(\mathrm{B})=\frac{3}{4}, \mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \overline{\mathrm{C}})=\frac{1}{3}, \mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B} \cap \overline{\mathrm{C}})=\frac{1}{3}$.
We know. $\mathrm{P}(\mathrm{B} \cap \overline{\mathrm{C}})=\mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \overline{\mathrm{C}})+\mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B} \cap \overline{\mathrm{C}})$
$=\frac{1}{3}+\frac{1}{3}=\frac{2}{3}$
$\mathrm{P}(\mathrm{B})=\mathrm{P}(\mathrm{B} \cap \mathrm{C})+\mathrm{P}(\mathrm{B} \cap \overline{\mathrm{C}})$
$\therefore \mathrm{P}(\mathrm{B} \cap \mathrm{C})=\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{B} \cap \overline{\mathrm{C}})$
$=\frac{3}{4}-\frac{2}{3}=\frac{9-8}{12}=\frac{1}{12} .$
We know. $\mathrm{P}(\mathrm{B} \cap \overline{\mathrm{C}})=\mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \overline{\mathrm{C}})+\mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B} \cap \overline{\mathrm{C}})$
$=\frac{1}{3}+\frac{1}{3}=\frac{2}{3}$
$\mathrm{P}(\mathrm{B})=\mathrm{P}(\mathrm{B} \cap \mathrm{C})+\mathrm{P}(\mathrm{B} \cap \overline{\mathrm{C}})$
$\therefore \mathrm{P}(\mathrm{B} \cap \mathrm{C})=\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{B} \cap \overline{\mathrm{C}})$
$=\frac{3}{4}-\frac{2}{3}=\frac{9-8}{12}=\frac{1}{12} .$
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