Question

For two events A and B, if $P\left(A\right)=P\left(A|B\right)=1/4$ and $P\left(B|A\right)=1/2$, then

• A. A and B are independent
• B. A and B are mutually exclusive
• C. $P\left(A^{\prime }|B\right)=3/4$
• D. $P\left(B^{\prime }|A^{\prime }\right)=1/2$

Answer 1

Answer: (A), (C), (D)

A and B are independent
$P\left(A^{\prime }|B\right)=3/4$
$P\left(B^{\prime }|A^{\prime }\right)=1/2$

We have
$P\left(A\right)=P\left(A|B\right)=\frac{P(A\cap B)}{P\left(B\right)}\Rightarrow P(A\cap B)=P\left(A\right)P\left(B\right)$
Therefore, A and B are independent. Also
$P(A\cap B)=P\left(A\right)P\left(B|A\right)=\left(1/4\right)\left(1/2\right)=1/8\ne 0$
$\therefore$ A and B cannot be mutually exclusive.
As A and B are independent
$P\left(A^{\prime }|B^{\prime }\right)=P\left(A^{\prime }\right)=1-P\left(A\right)=1-1/4=3/4$
Since A and B are independent,
$P\left(B\right)=P\left(B|A\right)=1/2$
$\Rightarrow P\left(B^{\prime }|A^{\prime }\right)=P\left(B^{\prime }\right)=1-P\left(B\right)=1/2$.