How likely is it that your vote will decide the election?
Well… that’s a difficult question to answer precisely. But there’s a similar question that is much easier to answer:
How likely is it that your vote will decide the election if the winner of the election is determined by popular vote, there are only two candidates, there are an odd number of total voters, and everyone but you votes by flipping a coin?
The answer to this question is exactly the probability of a binomial random variable taking the value . Stirling’s approximation yields:
Unsurprisingly, larger n (i.e. more voters) means an asymptotically 0 probability that you will decide the next election.
But before you decide to stay home, bear in mind that the amount of money at stake is also a function of n. In fact, considering that most taxes are defined in terms of a percentage of income or property value, it’s reasonable to assume that federal expenditures scale linearly with the size of the population.
So the expected value of voting is roughly:
In other words, the more people there are, the more valuable it is to vote.