Two balls are picked at random from an urn containing 5 red balls and 6 blue balls (without replacement). (a) What is the probability that the balls are the same color? (b) What is the probability at

Question

Two balls are picked at random from an urn containing 5 red balls and 6 blue balls (without replacement). (a) What is the probability that the balls are the same color? (b) What is the probability at

Answer 1

If two balls are selected without replacement, then we can consider this from a combinatorial perspective (permutations and combinations). First, let's consider all the ways 2 balls may be selected from a total of 11, without distinguishing them by color. In this kind of selection, order does not matter, so it is a combination.

The number of ways you can choose 2 balls from 11 is ₁₁C₂, which is 55. This is going to be the denominator.

Then we need to consider two cases: choosing 2 red balls, and choosing 2 blue balls. If both balls are red, then they were selected from the 5 red balls, which can happen in ₅C₂, or 10, ways. If both balls are blue, then they were selected from the 6 blue balls, which can happen in ₆C₂, or 15, ways.

Since success is considered choosing 2 red balls or 2 blue balls (and "or" typically means that we add), then we can add the fractions 10/55 + 15/55 = 25/55 = 5/11.