# Probability of choosing k pairs from n sets

If I have 3 bags of marbles containing an even number of marbles coloured say, 10 red in the first bag, 6 green in second bag, 4 blue in third. If I then draw marbles at random from any bag to make pairs without replacement so all pairs are enumerated (there will be 10 pairs in this case). What is the probability that I will end up with at least 2 pairs of red marbles?

And similarly how would I find the probability of obtaining at least one pair with one red and one green marble (the ordering is not important).

Ive been using a random sampling method to estimate this, but would be great if anyone could give me a hint as to how to solve this properly.

• So, to be clear: Each of three bags contains a set of 10:6:4 marbles, and you draw one pair from each bag? – Graham Kemp Nov 2 '16 at 21:23
• 10 marbles in one bag, 6 in another and 4 in the final bag. A pair is made by drawing from any bag. So a pair could be red-red or blue-red, etc – kezzos Nov 2 '16 at 21:32
• Then how many pairs are drawn, how are bags selected, and what happens if the selected bag is empty? – Graham Kemp Nov 2 '16 at 21:37
• Thanks for the help, there will be 10 pairs at the end of the experiment in this example. Bags are selected at random, if a bag is empty another bag must be selected. – kezzos Nov 2 '16 at 21:40