Winning an election by 1 vote. What are the chances?

December 20, 2017 By Alan J. Salzberg
Voting is thought of as the most important civic duty.  But does your one vote really count?  I mean, how many elections are won by just a vote?  Certainly not many.  Yet, I just read that in Virginia, an election for state representative was won by a single vote!  In this election there were a total of 23,215 votes and the final outcome (as of a recount and as of this writing) was 11,608-11,607.  That's right, it was won by a single vote.  All those voters who would've voted for the candidate with 11,607 votes but didn't bother to show up are really kicking themselves now.

To figure out the chances of winning by a vote, let's first make a couple of simplifying assumptions.  First, let's assume that the eligible voting age population is exactly split 50-50 in their preferences (so if everyone voted the election would be a tie).  For simplicity, let's also assume this population is much larger than the number of people who actually vote (this is sometimes true in smaller elections but less true in national elections).

If there are just 3 people voting, the chances of it being 2-1 or 1-2 are 75% (6 out of 8).  We can figure  this out by simply enumerating the 8 equally likely possibilities of the three people and counting how many are 1 vote wins (A is one candidate and B is the other):

AAA, AAB, ABA, BAA, BAB, BBA, ABB, BBB  : 6 of the 8 (all but the first and last) are a 2-1 win.  There is a faster way to do this by using a "binomial" distribution.  Using the R language, you can just double the chances of it being 2-1 (since 1-2 and 2-1 are equally likely) and the code looks like this: 2*dbinom(1,3,.5) .  But the R language and binomial distribution are for another time.

It is probably not surprising that as you increase the number of people voting, a 1 vote spread, even if the electorate is evenly split, is less and less likely.  For a coin toss, you might have heard the percent heads would get closer and closer to 50% as you flip more and more times.  That is true.  However, the *difference* in the number of heads and tails gets larger and larger.  The chance of a one vote difference is the same as the chance of there being a difference of only one between heads and tails, which gets small as the flips increases.

For 1,001 voters, the one vote spread has a chance of about 5%.

For 10,001 votes, the one vote spread has a chance of 1.6%.

For 23,215 votes (the number in the Virginia election), the chances are just 1.0%

For a presidential election with 100 million votes?  It's 0.016%, or about 1 in 6,000.  Pretty small, but certainly large enough for me to go to the ballot box.