Question

A box contains $b$ blue balls and $r$ red balls. A ball is drawn randomly from the box and is returned to the box with another ball of the same colour. The probability that the second ball drawn from the box is blue is

• A. $\frac{b}{r+b}$
• B. $\frac{b^2}{(r+b{)}^2}$
• C. $\frac{b+1}{r+b+1}$
• D. $\frac{b(b+1)}{(r+b)(r+b+1)}$

Answer 1

The correct option is A $\frac{b}{r+b}$

Case #1 If the first drawn ball is blue.

Total number of blue balls in box when returned with another blue ball$=b+1$

and the number of red balls is same.

then the probability that the second ball is blue is $P_1$

$P_1=\frac{b}{b+r}\times \frac{b+1}{r+b+1}$

Case #2 If the first drawn ball is red

Total number of red balls in box when returned with another red ball$=r+1$

and the number of blue balls is same.

then the probability that the second ball is blue is $P_2$

$P_2=\frac{r}{b+r}\times \frac{b}{r+b+1}$

Probability when second drawn ball is blue $=P=P_1+P_2=\frac{b(b+r+1)}{(b+r)(b+r+1)}=\frac{b}{b+r}$