Question

If A and B are two events such that $P\left(A\right)>0$ and $P\left(B\right)\ne 1$, then $P\left(\overline{A}/\overline{B}\right)=$

• A. $1-\frac{P(A\cup B)}{P\left(\overline{B}\right)}$
• B. $1-\frac{P(A\cup B)}{P\left(B\right)}$
• C. $\frac{1-P(A\cup B)}{1-P\left(B\right)}$
• D. $\frac{P(A\cup B)}{P\left(B\right)}$

Answer 1

The correct option is C $\frac{1-P(A\cup B)}{1-P\left(B\right)}$

$P\left(A^{\prime }/B^{\prime }\right)=\frac{P(A^{\prime }\cap B^{\prime })}{P\left(B^{\prime }\right)}$

According to De Morgan's Law: $P(A^{\prime }\cap B^{\prime })={P(A\cup B)}^{\prime }$

So $=P\left(A^{\prime }/B^{\prime }\right)=\frac{P(A\cup B{)}^{\prime }}{P\left(B^{\prime }\right)}$
$=\frac{1-P(A\cup B)}{1-P\left(B\right)}$