Question

If $A$ and $B$ are independent events such that $0<P\left(A\right)<1,0<P\left(B\right)<1$ then

• A. $A,B$ are mutually exclusive
• B. $A$ and $\overline{B}$ are independent
• C. $\overline{A},\overline{B}$ are independent
• D. $P\left(A/B\right)+P\left(\overline{A}/B\right)=1$

Answer 1

Answer: (B), (C), (D)

The correct options are
B $A$ and $\overline{B}$ are independent
C $\overline{A},\overline{B}$ are independent
D $P\left(A/B\right)+P\left(\overline{A}/B\right)=1$

If A and B are independent then $P(A\cap B)=P\left(A\right)P\left(B\right)nowP(A\cap \overline{B})=P\left(A\right)-P(A\cap B)=P\left(A\right)-P\left(A\right)P\left(B\right)=P\left(A\right)\{1-P\left(B\right)\}=P\left(A\right)P\left(\overline{B}\right)$
Hence A and $\overline{B}$ are independent.
similarly $P(\overline{A}\cap \overline{B})=1-P(A\cup B)=1-P\left(A\right)-P\left(B\right)+P(A\cap B)=1-P\left(A\right)-P\left(B\right)+P\left(A\right)P\left(B\right)=1-P\left(A\right)-P\left(B\right)\{1-P\left(A\right)\}=\{1-P\left(A\right)\}\{1-P\left(B\right)\}=P\left(\overline{A}\right)P\left(\overline{B}\right)$
Hence $\overline{A}$ and $\overline{B}$ are independent.
and $P\left(A/B\right)+P\left(\overline{A}/B\right)=\frac{P(A\cap B)}{P\left(B\right)}+\frac{P(\overline{A}\cap B)}{P\left(B\right)}=\frac{P\left(A\right)P\left(B\right)+P\left(\overline{A}\right)P\left(B\right)}{P\left(B\right)}=P\left(A\right)+P\left(\overline{A}\right)=1$
Option A is incorrect because $P(A\cap B)=P\left(A\right)P\left(B\right)\ne 0$ since A/Q $P\left(A\right)>0andP\left(B\right)>0$