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  4. If A and B are events of a random experiment such that P(A ∪ B) =(4/5) , P ( A ∪ barB) = (7/10) and P(B) = (2/5) , then P(A) equals

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Q. If $A$ and $B$ are events of a random experiment such that $P(A \cup B) =\frac{4}{5} , P ( A \cup \bar{B}) = \frac{7}{10}$ and $P(B) = \frac{2}{5} ,$ then $P(A)$ equals

BITSATBITSAT 2009

Solution:

Given, $P(\bar{A} \cup \bar{B} ) = P(\overline{A \cap B} ) = \frac{7}{10}$ ,
Since, $P(A \cap B) + P( \overline{A \cap B}) = \frac{1}{3}$
$\Rightarrow \, \, \, P \, \, A \cap B = 1 - \frac{7}{10} = \frac{3}{10} $
Also, $P(A \cap B) = P(A) + P(B) - P(A \cap B)$
$\Rightarrow \, \, \, \frac{4}{5} = P \, A + \frac{2}{5} - \frac{3}{10}$
$\Rightarrow \, \, \, P \, A = \frac{4}{5} - \frac{2}{5} + \frac{3}{10}$
$ = \frac{2}{5} + \frac{3}{10} = \frac{7}{10}$