Question

If $A$ and $B$ are independent events such that $0<P\left(A\right)<1,0<P\left(B\right)<1,$ then which of the following statement is/are correct ?

• A. $A$ and $B$ are mutually exclusive events.
• B. $A$ and $B^c$ are independent events.
• C. $A^c$ and $B^c$ are independent events.
• D. $P\left(A/B\right)+P\left(A^c/B\right)=1$

Answer 1

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

$P\left(A/B\right)+P\left(A^c/B\right)=1$

Given, $A$ and $B$ are independent events $i.e.$ occurrence of events does not depend on each other.
$\Rightarrow A^c$ and $B^c$ are also independent.
Similarly, $A$ and $B^c$ are also independent and vice-versa.
$\Rightarrow P\left(A/B\right)=P\left(A\right),P\left(B/A\right)=P\left(B\right),$ similarly
$P\left(A^c/B\right)=P\left(A^c\right).$
Since, $P\left(A\right)+P\left(A^c\right)=1$
$\Rightarrow P\left(A/B\right)+P\left(A^c/B\right)=1$