Question

A lot contains $50$ defective and $50$ non-defective bulbs. Two bulbs are drawn at random one at a time with replacement. The events $A,B,C$ are defined as :
$A:$ $1$st bulb is defective
$B:$ $2$nd bulb is non-defective
$C:$ both are defective or both are non-defective, then

• A. $A,B,C$ are pairwise as well as mutually independent.
• B. $A,B,C$ are pairwise independent but mutually not independent.
• C. $A,B,C$ are mutually independent but pairwise not independent.
• D. None of these.

Answer 1

The correct option is B $A,B,C$ are pairwise independent but mutually not independent.

$P\left(A\right)=\frac{50}{100}=\frac{1}{2}=P\left(B\right)$
$P\left(C\right)=\frac{1}{2}\cdot \frac{1}{2}+\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{2}$
$P(A\cap B)=P\left(A\right)\cdot P\left(B\right)=\frac{1}{4}$
$P(B\cap C)=P\left(B\right)\cdot P\left(C\right)=\frac{1}{4}P(A\cap C)=P\left(A\right)\cdot P\left(C\right)=\frac{1}{4}$
and $P(A\cap B\cap C)=0$
So, $A,B,C$ are pair wise independent but not all mutually independent.