A bag contain `n` ball out of which some balls are white. If probability that a bag contains exactly `i` white ball is pro0portional to `i^3dot` A ball is drawn at random from the bag and found to be white, then find the probability that bag contains exactly 2 white balls.

Explanation

Let evetn `A_(i)` is 'bag contains exactly I white balls'. According to the question, `P(A_(i))=ki^(2).`
`therefore underset(i=1)overset(n)sumki^(2)=1`
`thereforek=(6)/(n(n+1)(2n+1))`
`impliesP(A_(i))=(6i^(2))/(n(n+1)(2n+1))`
Let evetn B is "white ball is drawn" then using total probability theorem,
`P(B)=underset(i=1)overset(n)sumP(B//A_(i))P(A_(i))`
`=underset(i=1)overset(n)sum((i)/(n).(6i^(2))/(n(n+1)(2n+1)))`
`=(6)/(n^(2)(n+1)(2n+1))(n^(2)(n+1^(2)))/(4)`
`=(3(n+1))/(2(2n+1))`