Sex distribution 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 problem also gives the answer why you are "always" in the longer queue at the supermarket.
If $k=1$ the answer is trivial: All groups are gender balanced.
Therefore we shall assume that $k>1$.
Assume Samuel and Samantha were ill the day the groups were originally formed.
If the two Sams are assigned to the groups randomly, the result is just as if there had been no illness.
If Sam and Sam are asigned to the same group (which happens with probability $\frac1k$ if the distribution is uniform, with other distributions, your mileage may vary), we see by symmetry that more guys is exactly as probable as more gals for this group.
If Sam is assigned to a different group than Sam (which happens with probability $1-\frac1k>0$ in the uniform case, but we actually need only assume that the probability is $>0$), then three cases are possible:


*

*The group was gender-balanced before with some probability $p$, say (clearly, $p>0$, though the exact value depends on the group size).

*Or the group had more male members.

*Or the group had more female members. 


By symmetry, the probabilities for the latter two events are equal, hence are $\frac{1- p}2$ each. Then the probability that the group including Sam has more people of the same gender as Sam is at least $p+\frac{1- p}2=\frac{1+ p}2>\frac12$.
In total, it is more probable to find oneself in a group with more people of ones own gender than with less.
This holds both for Sam=Samual and Sam=Samantha.
