Question

Two events $A$ and $B$ will be independent if

• A. $A$ and $B$ are mutually exclusive
• B. $P(A^{\prime }\cap B^{\prime })=(1-P(A\left)\right)(1-P(B\left)\right)$
• C. $P\left(A\right)=P\left(B\right)$
• D. $P\left(A\right)+P\left(B\right)=1$

Answer 1

The correct option is B $P(A^{\prime }\cap B^{\prime })=(1-P(A\left)\right)(1-P(B\left)\right)$

For $A$ and $B$ to be independent,
$P(A\cap B)=P\left(A\right)P\left(B\right)$
Now,
$P(A^{\prime }\cap B^{\prime })=P(A\cup B{)}^{\prime }$
$=1-P(A\cup B)$
$=1-P\left(A\right)-P\left(B\right)+P(A\cap B)$
$=(1-P(A)-P(B\left)\right(1-P\left(A\right)\left)\right)$
$=(1-P(A\left)\right)(1-P(B\left)\right)$