Question

Suppose that there are two bags, each containing n − 1 red balls and one green ball. You choose k balls at random from the first bag and put them into the second. You then choose k balls from the second bag and put them into the first. Find the probability (in terms of n and k) that both green balls end up in the same bag.

Answer 1

Probability that both the green balls end up in the same bag

= Probability that both green balls end up in the second bag + Probability that both the green balls end up in the first bag

= Probability that in first k balls drawn, green ball is drawn too * Probability that in a total of (n + k) balls, when we draw k balls from second bag, no green ball is taken into the first bag + Probability that in first k balls drawn, green ball is drawn too * Probability that in a total of (n + k) balls, when we draw k balls from second bag, both green balls are taken into the first bag + Probability that in first k balls drawn no green ball is drawn * Probability that in (n + k) balls from second bag, we are drawing 1 green and (k-1) red balls

This is the required probability here.