Find Probability of X=3

Question

A bag had 1 red ball, 1 blue ball, and 12 green balls. Balls are taken out of the bag at random, one after the other and without replacement until a green ball is picked, at which point no more balls are picked. The random variable X represents the total number of balls taken out of the bag.

Find Pr(X=3)

Answer 1

At the start of the problem, there are 14 balls in the bag.

Only two of the balls are not green. Since we don't want to draw a green ball on the first try, we must draw either the red or blue ball. The probability of this is 2/14.

The same reasoning is used for the second draw. Of the 13 balls left in the bag, only one of them is not green. This is the ball that must be drawn, so the probability is 1/13.

On the third draw, every ball in the bag is green. In this case, you are guaranteed to draw a green ball. This means that the probability is 1.

The overall probability that X=3 is then (2/14) * (1/13) * 1 = 1/91.