Question

Two events $A$ and $B$ are said to be independent, if

• A. $A$ and $B$ are mututally 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

Answer: (B)

$P(A^{\prime }\cap B^{\prime })=[1-P(A\left)\right][1-P(B\left)\right]$

Two events A & B are independent if

$P(A\cap B)=P\left(A\right).P\left(B\right)$

Replacing A with A' and B with B'

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

As $P\left(A^{\prime }\right)=1-P\left(A\right)$

$P(A^{\prime }\cap B^{\prime })=[1-P(A\left)\right][1-P(B\left)\right]$