Question

A person is asked to randomly pick two balls from a bag which has 15 red and 5 blue balls. The probability that the person picks two balls of the same colour is

Answer 1

You pick 2 balls and they can be of the same colour if either both of them are blue or both are red, i.e. the effective probability(P) = P(RR)(both are red) + P(BB)(both are blue)

1: Both the balls are blue
The number of ways of picking up a blue ball out of 20 balls are 5/20. Number of ways of picking up a second blue ball are 4/19.
Thus probability of picking up 2 blue balls P(BB) = 5/20 * 4/19 = 20/380

2: Both the balls are red
The number of ways of picking up a red ball out of 20 balls are 15/20.
Number of ways of picking up a second red ball are 14/19.
Thus the probability of picking up 2 black balls P(RR) = 15/20* 14/19 = 210/380

Thus the effective probability P = 20/380+ 210/380 = 230/380= 23/38.

Answer 2

Total sample space = 20C2 = 190 The number of ways of selecting 2 balls of same color from a suit of 15 red balls 15C2 = 105 The number of ways of selecting 2 balls of same color from a suit of 5 blue balls 5C2 = 10 Thus the probability that the person picks 2 balls of same color = [105/190 + 10/190] = 115/190 = 23/38