The Probability of Close Elections

by Sanjeev Kulkarni

Department of Electrical Engineering

Princeton University

1. Some Close Races

The recent election resulted in a number of extremely close races where the margin of victory seemed surprisingly small. For example, in just the presidential race, consider the counts in the following states:

-- In Florida the original count resulted in a difference of only about 1700 votes between Bush and Gore with nearly 6 million votes cast. And this difference narrowed to less than 400 after a recount

-- In New Mexico, it appears that the race is within 100 votes out of about 600,000.

-- In Iowa, the difference between Gore and Bush is about 5,000 out of about 1.3 million votes cast.

-- In Wisconsin, the difference is about 6,000 out of about 2.5 million votes cast.

These seem like razor-thin margins, but are they really that surprising?

Moreover, surprising or not, the fact that the races are so close means that other unwanted influences (such as errors in vote counts) will be crucial in deciding our next president. In cases where such unwanted influences are of same order of magnitude (or greater) than the likely margin between the candidates, the election is really "in the noise" in the sense that the outcome may be decided by the effect of these unwanted (often random) influences. This appears to be the case in Florida, which in turn will determine the next president.

2. Chance of Getting of These Close Races

In a presidential race that numerous polls indicated was going to be a close one, some simple statistical calculations suggest that the narrow margins like those above may not be so surprising after all.

To calculate the probability of a close race, we need a model for voters. To simplify matters, let's suppose Bush and Gore are the only two candidates. The simplest model for a "close race" is that each voter simply "flips a coin" to decide for which candidate to vote. That is, each voter is equally likely to vote for either Bush or Gore, and their vote each is independent of all other votes. For this simple statistical model, one can compute the probabilities for each of the examples above. If we do this, the probabilities that the race is as close or closer than those in the examples above are as follows:

-- 13% chance with numbers as in Florida

-- 10% chance with numbers as in New Mexico

-- 99.9% chance with numbers as in Iowa

-- 99.9% chance with numbers as in Wisconsin

3. Effects of Third Party Candidates and Voter Turnout

It's natural to ask how third party candidates and voter turnout effect the results above. It turns out that a third candidate will only increase the probability of a close race between the two front runners. This is because the chance that the margin between two candidates will be large increases as the number of voters for the two candidates increases. A third candidate only takes away voters for the other two, increasing the probability of a tight race between the two front runners.

A little care needs to be taken in how we handle the issue of voter turnout. If the individuals who cast votes are chosen randomly from among all registered voters and we fix the total voter turnout and ask about the probability of getting a race within a certain margin, then we will get exactly the same answer as before. On the other hand, a more realistic model might be to assume that every registered person votes with a certain probability, chosen so that the average number of voters equals the actual turnout. In this case, the actual number of votes cast will be random. This complicates the computation, but will only slightly change the probabilities.

Even if we assume ALL registered individuals actually cast a vote, it won't change the basic point of the argument illustrated by the probabilities above. For example, suppose in Florida we assume 12 million people cast votes (for either Bush or Gore) instead of the 6 million assumed above. Then the probability of a margin of 400 or less is about 9% (instead of 13%). Likewise, in Wisconsin if we assume 5 million people cast votes (instead of 2.5 million assumed above), this will change the probability of a 6000 margin or less to about 99.2% instead of 99.9%. And these modest changes in the probabilities are are with rather drastic changes in the number of actual voters (i.e., doubling the number of voters).

In summary, neither of these two issues materially changes the numbers or the point of the argument. The calculation gets a little more cumbersome, but the change in the probabilities will be small. Moreover, the two effects together will tend to offset each other since they influence the probabilities in opposite ways. That is, increasing the number of (potential) voters may decrease the probability of a close race, while a third candidate increases this probability.

4. Some Final Comments

First, it should be emphasized that the probabilities for the tight races computed above assume the voters are undecided between the candidates. This may be a reasonable model for elections such as the recent one, but is clearly not a good model when there is a strong preference for one candidate. Nevertheless, given a tight race, the seemingly narrow margins are not as unlikely as they might seem.

What's surprising is that margins as close as those above are not observed more frequently! Of course, voters are not coin-flippers, and are not completely independent of one another. In fact, the lack of margins as narrow as those above in most races suggests that, as a whole, the American people are usually decidedly in favor of one candidate over the other.

What's also surprising, although perhaps disturbing is a better word, is the amount by which the vote count changed in Florida after a re-count. The margin changed by well over 1000 votes. This is more than twice the current difference between the two candidates. Moreover, under the assumptions above, there's a better than 30% chance that the difference between Bush and Gore would be 1000 votes or less with 6 million voters.

Thus, the errors made in counting votes are comparable to a likely margin between the candidates! And, this ignores the possibly thousands of other corrupted or uncounted ballots. In Florida, the election really is "in the noise."

At best, such errors are purely random, another coin-flip of sorts. However, given technology that's been available for years, and the precision with which voting could be carried out with modest effort, it's insane to have errors play such an important role in selecting our president. Whether Bush or Gore becomes our next president, there's a very good chance that the deciding factor came down to a coin flip (unwanted random influences) rather than the will of the American people.