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

If $A$ and $B$ are independent events, i.e., the occurrence of one does not affect the occurrence of the other, then obviously $\overline{A}$ and $\overline{B}$ are also independent of each other. Similarly, $A$ is independent of $\overline{B}$ and vice-versa.
Again, since $A$ and $B$ are independent events, $P\left(A|B\right)=P\left(A\right)$ and similarly, $P\left(\overline{A}|B\right)=P\left(\overline{A}\right)$.
Since $P\left(A\right)+P\left(\overline{A}\right)=1$, hence, $P\left(A|B\right)+P\left(\overline{A}|B\right)=1$.