If $P(A ∩ B)=\frac{1}{2}, P(\overline{A} ∩ \overline{B})=\frac{1}{2}$ and $2P(A)=P(B)=p$, then the value of p is equal to
Answer & explanation
Correct answer: option 2
We have,
$P(A ∩ B)=\frac{1}{2}, P(\overline{A} ∩ \overline{B})=\frac{1}{2}$ and $ 2P(A)=P(B)=p$
Now, $ P(\overline{A} ∩ \overline{B})=\frac{1}{2}$
$⇒ P(\overline{A ∪ B})=\frac{1}{2}$
$⇒ 1- P (A ∪ B) =\frac{1}{2}$
$⇒ P (A ∪ B) =\frac{1}{2}$
$⇒P(A) +P(B)- P(A ∩ B)=\frac{1}{2}$
$⇒ \frac{p}{2} +p -\frac{1}{2}=\frac{1}{2}⇒ \frac{3p}{2}=1⇒p=\frac{2}{3}$