$If\ A\ and\ B\ are\ independent\ events\ of\ a\ random\ experiments\ such\ that$ $P(A \cap B) = \frac{1}{6}and\ P\left( \overline{A} \cap \overline{B} \right) = \frac{1}{3},\ then\ P(A)\ is\ equal\ to$

Question

Dr. Abubacker · Accepted Answer

(b)
Since, $A\ and\ B\ are\ independent\ events$
$	herefore P(A)P(B) = \frac{1}{6}\ and\ P\left( \overline{A} ight)P\left( \overline{B} ight) = \frac{1}{3}$
$\Rightarrow \ \left\lbrack 1 - P(A) ightbrack\left\lbrack 1 - P(B) ightbrack = \frac{1}{3}$
$\Rightarrow \ 1 - \left\lbrack P(A) + P(B) ightbrack + P(A)P(B) = \frac{1}{3}$
$\Rightarrow \ 1 + \frac{1}{6} - \frac{1}{3} = P(A) + P(B)$
$\Rightarrow \ P(A) + P(B) = \frac{5}{6}$
$\Rightarrow \ P(A) = \frac{1}{2}\ ,\ P(B) = \frac{1}{3}\ ,\$
$or\ P(A) = \frac{1}{3}\ ,\ P(B) = \frac{1}{2}$

Q. $If\ A\ and\ B\ are\ independent\ events\ of\ a\ random\ experiments\ such\ that$ $P(A \cap B) = \frac{1}{6}and\ P\left( \overline{A} \cap \overline{B} \right) = \frac{1}{3},\ then\ P(A)\ is\ equal\ to$

Explanation

Reviewed by Dr. Abubacker

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