Question

$A$ & $B$ are two independent events such that $P\left(\overline{A}\right)=0.7,P\left(\overline{B}\right)=a$ & $P(A\cup B)=0.8$, then $a=$

• A. $5/7$
• B. $2/7$
• C. $1$
• D. None

Answer 1

Answer: (B)

$2/7$

Given, $P\left(A^{\prime }\right)=0.7$
But $P\left(A\right)=1-P\left(A^{\prime }\right)=1-0.7=0.3$
similarly $P\left(B^{\prime }\right)=a$
$\Rightarrow P\left(B\right)=1-a$
$P(A\cap B)=P\left(A\right).P\left(B\right)$ (since $A,B$ are independent events)
$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$
$\Rightarrow 0.8=0.3+1-a-(0.3\ast (1-a\left)\right)$
$\Rightarrow 0.8=0.3+1-a-0.3+0.3a$\Rightarrow 0.2=0.7a\Rightarrow a=2/7$