Question

A bag contains $n$ balls. It is given that the probability that among these n balls exactly $r$ balls are white is proportional to $r^2 (0 \le r \le n)$. A ball is drawn at random and is found to be white. Then the probability that all the balls in the bag are white, will be:

• A. $\frac{2n}{\left(n+1\right)^{2}}$
• B. $\frac{4n}{\left(n+1\right)^{2}}$
• C. $\frac{2n}{\left(n+3\right)^{2}}$
• D. $\frac{4n}{\left(n+3\right)^{2}}$

Answer 1

Answer: (B)

Let $E_i = 0, 1, 2, ... n$ be the event that the bag contains exactly i white balls then $p (E_i) k^2i$
where $\displaystyle \sum_{i=0}^n k^2_i=1$ $\Rightarrow \:k=\frac{r}{n\left(n+1\right)\left(2n+1\right)}$
Let A be the event that a ball drawn is white
$p\left(\frac{E_n}{A}\right)=\frac{p\left(E_n\right)p\left(A/E_n\right)}{\displaystyle \sum_{i=0}^nk^2_i\left(\frac{i}{n}\right)}$$=\frac{K.n^2.1}{\frac{k}{n}\displaystyle \sum_{t=0}^n(i)^3}$$=\frac{n^{3}}{\left(\frac{n\left(n+1\right)}{2}\right)^{2}}$
$p\left(\frac{E_{n}}{A}\right)=\frac{4n}{\left(n+1\right)^{2}}$