Lets say CA(where I live) was a swing state (it isn't) and very close, so close in fact that the odds of any given person voting Republican vs. Democratic are exactly 50%/50% (It won't be). Lets further say that the same number of people will vote as in 2008, 12,255,311 people (likely to be far more if CA actually was a swing state) - reference wikipedia. Consider each person's vote a binomial random variable with p = 0.5 (50%) of voting either way and n = 12,255,310 people (I reduced by one to make this even). What is the probability that excluding my vote, the final vote will be exactly half of the people in each direction? Note, this is the only case where my vote will "count". This is easy to calculate. The probability of getting exactly k successes in n trials is given by the probability mass function:
Which is a pain to evaluate, even MatLab kinda bombs (I get an answer, but has very obvious error). If you approximate this as a normal distribution, which is fine given the large N, then you get good results in MatLab (octave):
n = 12255310
octave:36> normpdf(n/2, n/2, n/4)
ans = 1.3021e-07
So, thats our answer. The probability of my vote affecting CA's outcome given our very generous inputs is .000013%. I think even this number is probably way high due to precision errors.
There is a philosophical approach to this as well that involves no math and is much shorter. Lets say that my vote is the deciding vote. If this were true, I sure as hell don't want to be that guy. What if I'm wrong? I am not going to spend the due diligence required to select the next leader of the free world. For one, none of the candidates are going to bother granting me an exclusive interview.
The obvious conclusion is that voting for Presidents is symbolic, not statistically meaningful. This is accurate in my humble opinion. However, you knew that. Yet you, like me, are still outraged about politics - either climate change (my favorite issue) or some other. Forget trying to elect a president. Your influence is much bigger in smaller ways. Work with local leaders: at your company, your neighborhood, your city, even maybe at a state level. Donate money to charities that are making a real difference (not lobbyists). Get involved. Changes can be made, but don't rely on a vote for a new president to make them happen.