Question

Let A and B be two events such that
P (A) = 0.5, P (B) = 0.3, and
$P\left(A|B\right)=0.5,$
Then $P\left(A^C|B^C\right)$ equals

• A. 0.5
• B. 0.6
• C. 0.15
• D. 0.3

Answer 1

The correct option is A 0.5

$\mathrm{Given}P\left(A\right)=0.5,P\left(B\right)=0.3\mathrm{and}P\left(A|B\right)=0.5\mathrm{We}\mathrm{know}\mathrm{that}P\left(A|B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}\therefore P\left(A\cap B\right)=P\left(A|B\right)\times P\left(B\right)=0.5\times 0.3=0.15$
$\mathrm{Now},P\left(A^C|B^C\right)=\frac{P\left(A^C\cap B^C\right)}{P\left(B^C\right)}=\frac{P{\left(A\cup B\right)}^C}{P\left(B^C\right)}=\frac{1-P(A\cup B)}{1-P\left(B\right)}=\frac{1-\left\{P\right(A)+P(B)-P(A\cap B\left)\right\}}{1-P\left(B\right)}=\frac{1-(0.5+0.3-0.15)}{0.7}=\frac{0.35}{0.7}=0.5$