Question

A bag contains $4$ balls out of which some balls are white . If probability that a bag contains exactly $i$ ball is proportional to $i^2$. A ball is drawn at random from the bag and found to be white, then the probability that bag conatins exactly 2 white balls is $p$ then $25p$ is

Answer 1

We have ,
$P\left(A_i\right)=k{i}^{2}1=k\sum n^2\Rightarrow k=\frac{1}{\sum n^2}\therefore P\left(A_i\right)=\frac{6i^2}{n(n+1)(2n+1)}$
Let event $B$ denote that the ball drawn is white. Then,
$P\left(B\right)=\frac{6}{n(n+1)(2n+1)}[1^2\frac{1}{n}+2^2\frac{2}{n}+...+n^2\frac{n}{n}]=\frac{3(n+1)}{2(2n+1)}P\left(A_2|B\right)=\frac{32}{\left[n\right(n+1){]}^2}$
here bag contains only $4$ balls
$\therefore n=4$
$\Rightarrow 25P\left(A_2|B\right)=2$