TITLE: What are Belgium's chances on getting a medal at the 400m finals, given that they have 2 of the 8 athletes?
QUESTION [26 upvotes]: This is a question my wife and I pondered over at the breakfast table this morning: Belgium has two athletes in the finals of the 400m at the World Championship athletics this year.
What are Belgium's chances of getting a medal, assuming every runner has the same capabilities? Plus, what's the formula/explanation for calculating this?
I know/think that, if we'd have 1 athlete, it would be 3/8, because there are 3 possible medals, and 8 athletes competing. (I hope this step is correct?)
But what if you have 2 athletes? Is it then just 6/8? Intuitively, that feels incorrect. But I would love to get a decent explanation on how to calculate this.

REPLY [4 votes]: There are $\binom83$ ways to pick three winners out of eight. If the Belgians do not win, there are $\binom63$ ways to pick three winners from the non-Belgians. Hence the probability that some Belgian would get a medel is $1-\binom63/\binom83 = 1-\frac{6!5!3!}{3!3!8!} = 1-\frac{5}{14}=\frac{9}{14}.$