Question

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 additional 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 does not depend on k.

Answer 1

n urn contain m white (W) and n Black (B) balls

Let (E1)= balls drawn is white in first draw

(E2)= balls drawn is Black in first draw

A drawing a White ball in 2nd draw

P(A)=P(E1) P(A/E1) + P(E2) P(A/E2)

P(E1)= m/(m+n)

P(E2)= n/(m+n)

Now if first ball is drawn is White'k' white ball are added Total balls are m+n+k

P(A/E1)= P(getting white in 2 draw when first ball drawn is'w')

P(A/E1) =({m+k}/{m+n+k})

P(A/E2)= P(getting white in 2 draw when first ball drawn is'b')

P(A/E2) =(m/{m+n+k})

P(A)=P(E1) P(A/E1) + P(E2) P(A/E2)

P(A) = m/(m+n) * ({m+k}/{m+n+k}) + n/(m+n) * (m/{m+n+k})

=(m(m+k) + mn )/ (m+n)(m+n+k)

=m(m+n+k) / (m+n)(m+n+k)

= m/(m+n)

this probability is independent of 'k'