# Probability of drawing without replacement at least one identical pair from two urns with n balls of n colors

Given two urns with an identical set of $$n$$ balls of $$n$$ different colors, where $$n$$ is even. After drawing without replacement $$n/2$$ pairs from the first urn and also drawing without replacement $$n/2$$ pairs from the second urn, what is the probability that at least one identical pair has been drawn? The order within a pair should not matter, so (A,B) is considered identical to (B,A).

• Welcome to Stats.SE. Take the opportunity to take the tour( stats.stackexchange.com/tour), if you haven't done it already. See also some tips on how to ask, on formatting help and on writing down equations using LaTeX / MathJax. If your question is clear and focused on your specific difficulty and you show your effort in solving the problem, it's more likely to get good and helping answers. – Ertxiem - reinstate Monica Sep 11 at 17:41
• What have you tried so far? Perhaps a similar problem is easier to solve: you may consider that each urn has $k$ identical balls. In this case, what is the probability of drawing one identical pair? – Ertxiem - reinstate Monica Sep 11 at 17:43
• Hi! Welcome here... and I agree with @Ertxiem's suggestions about better describing what the difficulty is. Anyway, it's a bit difficult to understand the question: how many draws are done from the second urn? I would assume one, but the observation again without replacement makes me wonder whether it's a different number (e.g. $n/2$ like from the first urn, or anything in between). – polettix Sep 11 at 17:47
• Are these pairs ordered or unordered? – whuber Sep 11 at 19:29
• Thanks for the input, I rephrased the question. – gnebehay Sep 12 at 7:01