Question

$A$ and $B$ are two events such that $P\left(A\right)=0.54,P\left(B\right)=0.69$ and P(A $\cap$B) $=0.35$
Find (i) P(A$\cup$B)
(ii) P(A' $\cap$B')
(iii) P(A $\cap$B')
(iv) P(B $\cap$A')

Answer 1

It is given that $P\left(A\right)=0.54,P\left(B\right)=0.69,P(A\cap B)=0.35$
(i) We know that $P$ $(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$
$\therefore P(A\cup B)=0.54+0.69-0.35=0.88$
(ii) $A^{\prime }\cap B^{\prime }={(A\cup B)}^{\prime },$ [by De Morgan's law ]
$\therefore P(A^{\prime }\cap B^{\prime })=P{(A\cup B)}^{\prime }=1-P(A\cup B)=1-0.88=0.12$
(iii) $P{(A\cap B)}^{\prime }=P\left(A\right)-P(A\cap B)=0.54-0.35=0.19$
(iv)We know that, $P(B\cap A^{\prime })=P\left(B\right)-P(A\cap B)$
$\therefore P(B\cap A^{\prime })=0.69-0.35=0.34$