# probability of event given bag of five pairs of balls

We are given a bag of 5 pairs of balls where balls in a pair have the same color, and different pairs have different colors. Five people pick two balls each from the bag, without replacement. What is the probability that all five people have a pair of balls with the same color?

I was thinking about the probability, for example, of the first person getting a pair, which I thought would be: $$\frac{5}{10 \choose 2}$$ And the probability of the second person getting a pair given the first person got a pair, which I thought would be: $$\frac{4}{8 \choose 2}$$ And for the third person (given the first and second got a pair): $$\frac{3}{6 \choose 2}$$ ... and so on for the fourth and fifth person. I'm not sure where to go from here, or if this is a right approach. This question is a practice problem for an introductory probability class, btw :)

As an alternate, possibly easier, method note that if there are $N$ pairs, the probability of drawing a pair is $\frac 1{2N-1}$ as the first draw can be anything, and then there is only one object left that works. Thus the answer here is $$\prod_{n=1}^5\frac 1{2n-1}=\frac 1{945}$$ You can verify that this matches your result.