Probability question

I have several acquaintances who don’t vote because ‘their one vote could never matter’.
I know that they all play the lottery, specifically PowerBall and MegaMillions.

My question is this: What are the odds of one vote making the difference in a presidential election? A gubernatorial election? Or any other state elections?

How does that compare to winning the aforementioned lotteries?

I realize the odds on winning the lottery are known, but is it even possible to answer the important part of the question?


Elections aren’t strictly probabilistic, so there’s no good answer to this question without adding enough assumptions to make the answer meaningless for general elections.

Also, for any election larger than a few thousand people, there’s enough noise that we don’t necessarily have confidence the results are totally valid down to the single vote, anyway (recounts often tally up differently than the official results).

As for the argument that one vote doesn’t matter, it certainly doesn’t matter a lot when compared against the aggregate, but it clearly matters, because you don’t have a reasonable aggregate unless sufficient people vote, i.e. if everybody bought into this argument, suddenly your single vote would clearly matter a great deal.

One thing you can say is that your single vote doesn’t matter enough to affect the aggregate, as you are reasonably assured of a reasonable turnout and especially close results. That’s usually true but not always true. For example, the 2000 Presidential elections could reasonably be said to hinge on Florida. Even then, a single vote wouldn’t have swung the tide, but, in aggregate, were there sufficient Nader voters who may have voted differently (or not at all) had they realized how close the result would have been? There’s certainly room to argue that point, since the result hinged on a few thousand votes.

And at that point, you have a pile problem. A single vote isn’t a sufficient aggregate. 10 votes probably isn’t, either. A million votes certainly is. 10000 votes is probably big enough, too. What about 50? Or 100? Or 1000? The problem is magnified when we have different levels in voting. Presidential elections depend on the individual states, and states have different numbers of electors. So, in a sense, a voter in Alaska has more influence than one in California. But California, in aggregate, has more influence than Alaska.

The recent election for Montana governor was decided by about 7600 votes. I think a reasonable estimate of the a priori chance that one new vote would affect that outcome would be very roughly about 1 in 15000, or 1 in twice the margin. (Someone who doesn’t understand the difference between prediction and postdiction will show up and volunteer that the chance is zero if the winning margin is greater than 1.)

Obviously this is much larger than the chance of winning a major lottery prize. On the other hand, if your vote cost $1 (or the inconvenience and gasoline of voting represents as little as $1), the election outcome would need to make $15,000 difference to you, to justify voting economically. This fact is yet another example of a “free-rider” problem, and is why some countries make voting mandatory.

Estimating the chance of determining a Presidential election is more difficult, but roughly the chance is zero unless you’re voting in a swing state, like Ohio or Florida.

xkcd analyzed a similar question recently

See Utilitarianism.

Why 1 in twice the margin?

If you vote for one, you don’t for the other? First thought…

I was thinking you need to move the difference from +1 to zero and win the coin-toss, but of course there’s also the case where you move from 0 to -1 and would have lost the coin-toss.

There are other flaws – I was just giving a very crude estimate.

In that article, there’s a link to a paper asking the following question: given the polling data 11 days before the 2008 election, what is the chance that your vote would decide the national election? In other words, what was the possibility that your vote would be decisive for your state, and that your state would have been necessary for the winner to get a majority in the electoral college?

The answer they obtained was greater than one in 10 million for a few swing states (CO, NH, VA, NM) and greater than one in 100 million for a bunch more (NV, ND, MN, IN, OH, FL, PA, MI, NC, MO, ME, and possibly WI, IA, and MT — it’s hard to tell from their graphs.) The average US voter had a one in 60 million chance of being decisive. For contrast, the probability of winning the jackpot on Mega Millions is about one in 259 million, and the probability of winning “only” $1 million is about one in 18 million.

This calculation is, of course, specific to that one election. Your chances would be much lower in a landslide year, and much higher in a hotly contested year. Still, it gives us a rough order of magnitude.

The 2008 movie Swing Vote had the Presidential Election come down to just one vote (better yet, the specific vote of a man played by Kevin Costner). They made it seem like it could really happen.

From the Wikipedia article on Swing Vote:

But if it came down to Kevin Costner’s vote, then it also came down to a bunch of other peoples’ votes, too.

I remember reading, many years ago, a short story, in which some future American presidential election had been reduced by computer analysis to the vote of one carefully selected individual. The story was how it affected him.

Ring a bell with anyone?

Isaac Asimov, “Franchise”


Some estimation of the theoretical odds can be made, using lots of assumptions.

Practically speaking, however, the chances are 0. For one thing, the universe of national elections is ridiculously low. There have only been around 100 presidential election. Even adding in all state elections doesn’t push the total much past 10,000. If the actual odds are hundreds of millions to one then it should be geologic ages before you see an occurrence.

An even larger barrier is the margin of error within any election. Close elections are routinely recounted. There has never, to my knowledge, been a major election recount in the which the final count varied from the original by just one vote. (It does happen in small town elections, but with no more than a few hundred votes.) Until we find a failproof system for recording votes, as well as determining who is eligible to vote, the uncertainties of the system will always be larger than 1.