Question

If $A$ and $B$ are two independent events, then the probability of occurrence of atleast one of $A$ and $B$ is given by

• A. $1-P(A)P(B)$
• B. $1-P(A)P(B^{\prime })$
• C. $1-P(A^{\prime })P(B^{\prime })$
• D. $1-P(A^{\prime })P(B)$

Answer 1

Answer: (C)

$P$(atleast one of $A$ and $B$)
$=P(A\cup B)=P(A)+P(B)-P(A\cap B)$
$=P(A)+P(B)-P(A)P(B)$ [$\because A$, $B$ are independent]
$=P(A)+P(B)[1-P(A)]=[1-P(A^{\prime })]+P(B)P(A^{\prime })$
$=1-P(A^{\prime })+P(B)P(A^{\prime })=1-P(A^{\prime })[1-P(B)]$
$=1-P(A^{\prime })P(B^{\prime })$