Your Vote Counts More Than You Think

Let’s dispel the myth that a single vote does not mathematically make a difference in an election.

Photo by Arnaud Jaegers on Unsplash

If a million ballots are cast in an evenly poised election between two candidates, what are the odds that your single vote could decide the election? Almost 0? 1 in 1,000,000? Well, actually it’s close to 1 in 1,000! Admittedly it’s still a small likelihood, but not as small as you thought, right? If someone gave me a 1 in 1,000 shot at directly choosing an elected official, and all I had to do was to cast a vote, I would take that in a heartbeat. Wouldn’t you?

The mathematical probability of a tie in an election between two evenly poised candidates is approximately 1 / √N, where N is the number of people who vote. We derive this result later. But first, consider the difference your vote can make in the upcoming 2020 US presidential election in toss-up states (using the Wikipedia data on the per-state total ballots cast in the 2016 presidential election and anticipated toss-up states from FiveThirtyEight).

║ State ║ Odds of a tie ║
║ Iowa ║ 1 in 1,564 ║
║ Arizona ║ 1 in 2,017 ║
║ Wisconsin ║ 1 in 2,156 ║
║ Georgia ║ 1 in 2,544 ║
║ North Carolina ║ 1 in 2,722 ║
║ Ohio ║ 1 in 2,931 ║
║ Pennsylvania ║ 1 in 3,104 ║
║ Texas ║ 1 in 3,744 ║
║ Florida ║ 1 in 3,837 ║

How can the chances of a tie not be vanishingly small when so many ballots are cast? Here’s the math (also see Andy Magid’s post on this):

Let’s unpack the math above. Consider the case of a small election with four people casting votes (N = 4) for candidates A and B, each of whom is equally likely to win. If each person votes for B, except the third person who votes for A, we could represent that choice as BBAB. Since each person can vote for either A or B, there are 2⁴ or 16 possible ways for the people to vote: AAAA, AAAB, AABA, AABB, ABAA, ABAB, ABBA, ABBB, BAAA, BAAB, BABA, BABB, BBAA, BBAB, BBBA, BBBB. Of the 16 possible choices, the number of choices in which the result is a tie is equal to the number of ways in which we can pick 2 persons out of the 4 people (which is 6): 1st&2nd, 1st&3rd, 1st&4th, 2nd&3rd, 2nd&4th, 3rd&4th. The picked persons cast their vote for A and the unpicked persons cast their vote for B. The case of the picked persons voting for B is already accounted for by the corresponding“complementary” pick i.e. if we pick the 1st and 3rd persons, then in the complementary pick of the 2nd and 4th persons, the 1st and 3rd vote for B. So the 6 ways in which the people’s vote results in a tie are AABB, ABAB, ABBA, BAAB, BABA, BBAA. And the probability of a tie is 6 / 16 or 3 in 8.

For a large number of voters (e.g. N = 1 million), a brute force enumeration of possibilities is prohibitive. Even simply computing the result from the exact formula is difficult due to the large numbers involved (e.g. the 2¹⁰⁰⁰⁰⁰⁰ in the denominator). It’s easier to use the Stirling approximation for factorials of large numbers, resulting in the insightful result that the probability of a tie is approximately 1 / N.

So please do vote in this election and in all future elections. Your vote could make all the difference!

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