Question

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

• A. (a) A and B are independent
  
• B. (b) A and B are mutually exclusive
  
• C. (c) P(A' | B ') =3/4
  
• D. (d) P(B' | A') =1/2

Answer 1

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

(d) P(B' | A') =1/2

We have,
P(A) =P(A|B) =$\frac{P(A\cap B)}{P\left(B\right)}\Rightarrow P(A\cap B)=P\left(A\right)P\left(B\right)$
Therefore, A and B are independent.
Since
$P(A\cap B)=P\left(A\right)P\left(B|A\right)=\left(1/4\right)\left(1/2\right)=1/8\ne 0$
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,
$P\left(B\right)=P\left(B|A\right)=1/2\Rightarrow P\left(B^{\prime }|A^{\prime }\right)=P\left(B^{\prime }\right)=1-P\left(B\right)=1/2$