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Gender distribution probability

Suppose there are N male and N female students. They are randomly distributed into k groups.

Is it more probable for a male student to find himself in a group with more guys and for female student to find herself in a group with more girls?

The question is motivated by an argument with my mother. She claimed that in the majority of collectives where she was, the number of women besides her was greater than men, while I had the opposite impression, that in most collectives where I was a member there was more men even if not to count myself.

I never was in a majority-girls collective, so I think for a male it is more probable to find oneself in a majority-male collective (even if we exclude oneself).

[This question on Stackexchange](https://math.stackexchange.com/questions/137568/sex-distribution)