I badly misread this problem:
You just won 2016 socks. Some of them are white, some are blue. The color of each sock was randomly chosen, with a 50/50 probability. Is it more probable that the socks can be paired, or that you will remain with two unmatching socks?
The correct answer is 50%, but I just woke up so I had this chain of thought:
That cannot be right. It's possible that all the socks might be blue, or all white, and then you cannot pair up any of them.
After my coffee, I realized that a "pair" is two socks of the same color, not, as my sleep-addled mind had insisted, one blue and one white.
But imagine, you are an eccentric who always wears one blue sock and one white. Now what are the odds of "pairability"?
For a more realistic example, consider a club at which men want to dance with women and vice-versa. The doorman admits the first 2016 people, without regard to who has already been admitted. Given that people show up randomly but in exactly equal proportions, What are the odds that exactly 1008 men and 1008 women are admitted?
(I am still pretty sleepy so my first instinct was to post on workplace.stackexchange.com for advice about hiring a better doorman. Then I came up with a solution that (a) isn't very good and (b) might not even be correct, given that 10 minutes ago, I was too sleepy to remember that socks are supposed be the same color. I will post it as an answer below, but feel free to critique or correct it.)