Question

An urn $A$ contains $2$ white and $3$ black balls. Another urn $B$ contains $3$ white and $4$ black balls.Out of these two urns, one is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that
I. It is from urn $A$ is $21/40$,
II. It is from urn $B$ is $20/41$
Which of the following statements is correct

• A. I only
• B. II only
• C. both I and II
• D. neither I nor II

Answer 1

Answer: (B)

II only

$P\left(A|B\right)=P(A\cap B)/P\left(B\right)$
Here $B=$getting black ball, $A=$ ball is from urn $A$
Hence, $P\left(A|B\right)=\frac{1}{2}\times \frac{\frac{3}{5}}{(1/2\times 3/5+1/2\times 4/7)}=21/41$
$P\left(\overline{A}|B\right)=1-P\left(A|B\right)$
$\Rightarrow P\left(\overline{A}|B\right)=1-P\left(A|B\right)$ $=1-21/40=20/41$