An urn contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k balls of the same colour as that of the ball drawn. a ball is again drawn at random. Show that the probability of drawing a white ball now does not depend on k.

Explanation

Let `W_(1)(B_(1))` be the event that a white (a black) ball is drawn in the first draw and let W be the event that white ball is drawn in the second draw. Then
`P(W)=P(B_(1))P(W//B_(1))+P(W_(1))P(W//W_(1))`
`=(n)/(m+nm+n+k)+(m)/(m+n)(m+k)/(m+n+k) `
`=(m(n+_m+k))/((m+n)(m+n+k))=(m)/(m+n)`