I would say that each way people vote are not equally likely--hence my original model of choosing 12002 voters randomly from a pool that was already evenly split between two candidates--which seems to be close to reality. If it were something like 25% Obama 75% Hillary overall, the 6001-6001 split would be far less likely. And a 3001-9001 split would be far more likely.
Of course, as I said, it all depends on how you model it--a true model would consider neurons in people, their likelihood of having an ancestor moving to Syracuse in the past few centuries, weather patterns, etc. etc....one reason math whizzes get hung up on probability (and even some famous ones have screwed up in the past....) is probability questions aren't generally stated mathematically, but you have to find a mathematical model that matches the question somewhat, making assumptions not stated in the question, etc. It's an art.
Tangential: I recall an argument between my high school Trig teacher (we had a probability month during the Trig year, as well as a few other random subjects since Trig doesn't require the full year) and myself--I still think I'm right, and I bet he still thinks he's right
Here is how it went.
You have a keychain. You put 5 different keys on it. How many ways can you do it? (really, a combinatorics question, but same kind of deal).
He said: doesn't matter how you put the first key on (by symmetry). 4 ways to choose the next key in line, 3 for the next, 2 for the next, 1 for the last, total: 4*3*2*1=24.
I said: doesn't matter how you put the key on (by symmetry). 4 for the next key, 3, then 2, then 1, just as before, but now if you had put the keys on in the reverse order, it would be the same arrangement! (just flip the keychain around). Every arrangement has a corresponding symmetric arrangement, so you divide by two, giving 12 possibilities. (so I got a grade of 99 on that test as a result.)