N1995 P2 Q6

A bag contains \(3\) red balls and \(3\) green balls. Balls are drawn from the bag at random, one by one and without replacement.

1. Show that the probability that the first \(3\) balls drawn are red is \(\frac{1}{20}\).[1]
2. Find the probability that the first \(3\) balls drawn consist of \(2\) red balls and \(1\) green ball (in an order). Hence, or otherwise show that the probability that the third red appear on the fourth draw is \(\frac{3}{20}\).[2]
3. Find the probability that the third red ball appears on the fifth draw.[2]
4. The random variable \(X\) is the number of draws required up to and including the one on which the third red ball appears. Tabulate the probability distribution of \(X\) and find \(\mathrm{E}\left( X \right)\).[4]