GRE probability question: What is the probability that the two marbles drawn will be the same color?

Question

Question: Bag A contains 3 white and 3 red marbles. Bag B contains 6 white and 3 red marbles. One of the two bags will be chosen at random, and then two marbles will be drawn from that bag at random without replacement. What is the probability that the two marbles drawn will be the same color?

Solution: 1) Choose from bag A: $\frac {_3C_2+_3C_2}{_6C_2} = \frac 25$. 2) choose from bag B: $\frac {_6C_2+_3C_2}{_9C_2} = \frac 12$. Since choosing A or B is random, multply both cases by $\frac 12$. Then, $\frac 15+\frac 14 = \frac 9{20}$.

I don't know how to establish $\frac {_3C_2+_3C_2}{_6C_2}$. Could you give some explanation about this?

Thank you in advance.

Answer 1

Explanation for the last line in question:

1. Drawing two same colour marbles can be done by choosing two red or two white marbles from a total of 3 each. Therefore it yields $3C2+ 3C2 $. (As work can be accomplished either by taking out two red or two white marbles.)
2. Total cases are $6C2 $as we are asked to take out two marbles from total of 6 marbles (red + white).