Let X denote the number of red balls in the outcome.

Here, x₁ = 2, x₂ = 1 , x₃ = 1 , x₄ = 0

Now, the probability of getting 2 red balls when we draw out the balls one at a time is:

Probability of first ball being red = 4/10

Probability of second ball being red = 3/9 (because there are 3 red balls left in the urn, out of a total of 9 balls left.) So:

Likewise, for the probability of red first is 4/10 followed by black is 6/9 (because there are 6 black balls still in the urn and 9 balls all together). So:

Similarly for black then red:

Finally, for 2 black balls:

As a check, if we have found all the probabilities, then they should add up to 1.

2/15 + 4/15 + 4/15 + 1/3 = 15/15 = 1

So we have found them all.