What Is the Probability of Exactly Two Matching Cards in Envelopes?

[PhysViti]

Homework Statement

Suppose that 10 cards, of which 5 are red and 5 are green, are placed at random in 10 envelopes, of which 5 are red and 5 are green. Determine the probability that EXACTLY two envelopes will contain a card with a matching color.

Homework Equations

The Attempt at a Solution

I know that the size of the sample space is 14C5 X 14C5. I know that there are 5X5=25 ways to select two envelopes of different colors. If there is only one red card in the red envelope, then the other red cards must be located in the green envelopes (since the problem specifies "exactly") with 8C4 possible ways. For all the cases:

1 red: 8C4
2 red: 7C3
3 red: 6C2
4 red: 5C1
5 red: 1

Therefore, there are 8C4+7C3+6C2+5C1+1 ways to satisfy the red card in red envelope condition. The same number of ways holds for the green card in the green envelope condition.

So, in all, there are 25 X (8C4+7C3+6C2+5C1+1)^2 events in the sample space that satisfy the problem's constraints. So, the probability is just the latter number over (14C5)^2.

How wrong is my answer?

 

[LCKurtz]

I think I would look at the problem this way. Just take 5 red envelopes. Selecting 5 cards at random from the 10, what is the probability of getting exactly 1 red? Looks like a hypergeometric distribution to me.