Question

There are two urns. There are $m$ white & $n$ black balls in the first urn and $p$ white & $q$ black balls in the second urn. One ball is taken from the first urn & placed into the second. Now, the probability of drawing a white ball from the second urn is

• A. $\frac{pm+(p+1)n}{(m+n)(p+q+1)}$
• B. $\frac{(p+1)m+pn}{(m+n)(p+q+1)}$
• C. $\frac{qm+(q+1)n}{(m+n)(p+q+1)}$
• D. $\frac{(q+1)m+qn}{(m+n)(p+q+1)}$

Answer 1

The correct option is B $\frac{(p+1)m+pn}{(m+n)(p+q+1)}$

Given:

Urn 1: $U_1$ has m white and n black balls.

Urn 2: $U_2$ has p white and q black balls.

The probability of white ball in urn 1 is $\frac{m}{m+n}$ and of black ball is $\frac{n}{m+n}$

Similarly, the probability of white ball in urn 2 is $\frac{p}{p+q}$ and of a black ball is $\frac{q}{p+q}$

One ball is taken from the first urn & placed into the second.

Now,

Case (i): If the transferred ball is white,

Urn 2: $U_2$ has (p+1) white and q black balls.

The probability of white ball in urn 2 is $\frac{p+1}{p+q+1}$ and of a black ball is $\frac{q}{p+q+1}$

Case (ii): If the transferred ball is black,

Urn 2: $U_2$ has p white and (q+1) black balls.

The probability of white ball in urn 2 is $\frac{p}{p+q+1}$ and of a black ball is $\frac{q+1}{p+q+1}$

Hence, the probability of drawing a white ball from the second urn is

$\frac{m}{m+n}\times \frac{p+1}{p+q+1}+\frac{n}{m+n}\times \frac{p}{p+q+1}⟹\frac{m(p+1)+np}{(m+n)(p+q+1)}$