Probability of choosing two marbles of the same color?

Question

A bag contains 4 red marbles, 5 yellow marbles, and 6 blue marbles. Three marbles are to be picked out randomly (without replacement). What is the probability that exactly two of them have the same color?

Answer 1

It seems easier to go via the complementary probability. There are ${4\choose3}+{5\choose3}+{6\choose3}=34$ ways to pick three equal marbles, and $4\cdot5\cdot 6=120$ ways to pick three different marbles. It follows that with probability $154/{15\choose3}={22\over65}$ we do not succeed in picking exactly two equal marbles. The requested probability then is ${43\over65}$.

Answer 2

Let's consider p the probability you're looking for p = p1 + p2 + p3 where: p1=p(choose exactly two red marbles) , p2 = p(choose exactly yellow marbles) and p3 = p(choose exactly two blue marbles). In order to calculate p1,p2 and p3, use the hypergeometric distribution and you get the answer.