Computing probabilities for each value of a random variable Assume you have a bag of 6 marbles, 2 of which are red and 4 of which are blue, and let the random variable Y represent the number of red marbles drawn after 3 are taken from the bag.
I'm trying to compute the probability for each value of Y.
What I've attempted:
Total number of combinations: ${}_6 \mathrm{ C }_3 = {6 \choose 3} = \frac{6!}{3!(6-3)!} = 20$
$P(Y = 0) = \frac{4 \choose 3}{20} = \frac{4}{20}$ (because there are four blue marbles in the bag and we are looking for the combinations of drawing three of these four)
$P(Y = 1) = 2\times\frac{4 \choose 2}{20} = \frac{12}{20}$ (because if you draw one red marble you have to draw two blue marbles from the bag to get Y = 1)
$P(Y = 2) = \frac{4 \choose 1}{20} = \frac{4}{20}$ (because if you draw two red marbles you have to draw one blue marble from the four in the bag to get Y = 2)
Have I done this correctly? I would appreciate any help.

I have a slightly different way of doing it, and my results are the same as yours.
$P(Y=0)=\frac{4\times 3\times 2}{6\times 5\times 4}=\frac{1}{5}$
$P(Y=1)=3\times \frac{2\times 4\times 3}{6\times 5\times 4}=\frac{3}{5}$
$P(Y=2)=3\times \frac{2\times 1\times 4}{6\times 5\times 4}=\frac{1}{5}$
