Question

Choose the correct answer,
Two events A and B are said to be independent, if
(a) A and B are mutually exclusive
(b) $P(A^{\prime }\cap B^{\prime })=[I-P(A\left)\right][1-P(B\left)\right]$
(c) $P\left(A\right)=P\left(B\right)$
(d) $P\left(A\right)+P\left(B\right)=1$

Answer 1

A and B are independent if $P(A\cap B)=P\left(A\right)P\left(B\right)$
$\therefore P(A^{\prime }\cap B^{\prime })=P(A\cup B{)}^{\prime }=1-P(A\cup B)=1-[P\left(A\right)+P\left(B\right)-P(A\cap B)]$
$[\because P(A\cup B{)}^{\prime }=1-P(A\cup B)]=1-P(A)-P(B)+P(A\left)P\right(B)=[1-P\left(A\right)\left]\right[1-P\left(B\right)]$