If $P(A) = P(B) = x\ and\ P(A \cap B) = P\left( A^{'} \cap B^{'} \right) = \frac{1}{3}$,then $x$ is equal to

Question

Dr. Abubacker · Accepted Answer

(a)
Given, $P(A) = P(B) = x$
and $P(A \cap B) = P\left( A^{'} \cap B^{'} ight) = \frac{1}{3}$
$	herefore\ P\left( A^{'} \cap B^{'} ight) = 1 - P(A \cup B)$
$\Rightarrow \ P(A \cup B) = 1 - \frac{1}{3} = \frac{2}{3}$ …(i)
Also, $P(A \cup B) = P(A) + P(B) - P(A) \cap B)$
$\Rightarrow \ \frac{2}{3} = 2x - \frac{1}{3}$ [from Eq (i)]
$\Rightarrow \ x = \frac{1}{2}$

Q. If $P(A) = P(B) = x\ and\ P(A \cap B) = P\left( A^{'} \cap B^{'} \right) = \frac{1}{3}$,then $x$ is equal to

Explanation

Reviewed by Dr. Abubacker

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