Question

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

• A. $P\left(A^{\prime }{B}^{\prime }\right)=[1-P(A\left)\right][1-P(B\left)\right]$
• B. $A$ and $B$ are mutually exclusive
• 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 $A$ and $B$ are mutually exclusive

We know that,

If $A\&B$ are independent events, then
$P(A\cap B)=P\left(A\right)\times P\left(B\right)$ .....(1)

Replacing $A$ as $A^{\prime }\&B^{\prime }$ in (1), then $P(A^{\prime }\cap B^{\prime })=P\left(A^{\prime }\right)\times P{B}^{\prime }$

Since,
$P\left(A^{\prime }\right)=1-P\left(A\right)\&P\left(B^{\prime }\right)=1-P\left(B\right)$
Therefore,
$P(A^{\prime }\cap B^{\prime })=[1-P(A\left)\right]\times [1-P(B\left)\right]$

Thus, option $A$ is correct answer.