Question

For any two events A and B, the conditional probability $P\left(B/A\right)=\frac{P(B\cap A)}{P\left(A\right)}$ and ifAand B are independent
$P(B\cap A)=P\left(B\right).P\left(A\right)$ So, $P\left(B/A\right)=P\left(B\right)$
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 : 1st bulb is defective
B : 2nd bulb is non-defective
C : both are defective or both are non-defective
then,

• A. A, B, C are pair-wise independent as well as mutually independent
• B. A, B, C are pair-wise independent but mutually not.
• C. A, B, C are mutually independent but pair-wise not
• D. none of these

Answer 1

Answer: (B)

A, B, C are pair-wise independent but mutually not.

$P\left(A\right)=50/100=1/2$
$P\left(B\right)=50/100=1/2$
$P\left(C\right)=\left(1/2\right)\left(1/2\right)+\left(1/2\right)\left(1/2\right)=1/2$
$P(A\cap B)=P\left(A\right).P\left(B\right)=1/4$
$P(B\cap C)=P\left(B\right).P\left(C\right)=1/4$
$P(A\cap C)=P\left(A\right).P\left(C\right)=1/4$
$P(A\cap B\cap C)=0$