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 $p(E_i)k^{2}i$
where $\sum _{i=0}^n{k}_i^2=1$ $\Rightarrow k=\frac{r}{n\left(n+1\right)\left(2n+1\right)}$
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)}{\sum _{i=0}^n{k}_i^2\left(\frac{i}{n}\right)}$$=\frac{K.n^2.1}{\frac{k}{n}\sum _{t=0}^n(i{)}^3}$$=\frac{n^3}{{\left(\frac{n\left(n+1\right)}{2}\right)}^2}$
$p\left(\frac{E_n}{A}\right)=\frac{4n}{{\left(n+1\right)}^2}$