This is a first in a series of posts looking at issues in the current political landscape where a better understanding of some basic math principles would likely change the direction of the discussion. We are an innumerate nation (the mathematical equivalent of illiterate). When we make political decisions based on bad math we are harming our fellow citizens, and often the world in which we live. And so, this is my modest attempt to nudge the conversation into a more productive direction.
The first topic, chosen because it is so easily debunked, is the longstanding Republican quest, funded by millions of dollars of lobbyist money and thousands of hours of legislative time, chasing a ghost called “voter fraud.” The idea is that many important elections, including the last several presidential elections, have been decided or threatened by masses of “illegal” voters (most assumed to be Hispanic). In short, the math does not hold.
The current Republican candidate for governor of Kansas, Kris Kobach, has made a well-financed career asserting horror stories of illegal voting. He even chaired President Trump’s ill-fated Commission on Election Integrity, which disbanded in early 2018 without filing a report, finding nothing beyond a few anecdotal cases of “registration fraud” or “impersonation fraud,” and the small number found had no visible partisan bias.
There are some “grand conspiracy” claims here that I will deal with in a later post, but first, we can debunk this myth with some basic questions:
What is the “natural error” in election vote counting?
Higher than you probably think. Most people are under the assumption that there is “one correct count” in elections. Once you get into the tens of thousands of votes, however, this is rarely true. First there is pure counting error, with studies finding that multiple recounts usually deliver a slightly different answer each time. Add to that unclear ballots, provisional ballots, “hanging chads” and other anomalies, and the final vote count “natural error” can be large.
Note here that provisional ballot issues are often politician-caused. People move, some quite frequently, but that should not invalidate one’s right to vote. Polling places and precinct boundaries also change often, confusing voters. Absentee ballot procedures and signature requirements can also be confusing, forcing otherwise valid ballots into provisional limbo, and some states have been over-aggressive in purging voter rolls.
One study found this “natural error” to range as high as 0.5% to 2% depending on the recount method. The error is reduced in the final impact on most elections because these are usually “unbiased” errors, offsetting the votes for one candidate versus another. 
What is the probability of a voter casting an illegal ballot?
Small. Really small. There are two primary types of alleged voter fraud. First, that non-citizens or convicted felons are illegally registering to vote in large numbers. Second, that people are impersonating legally-registered voters at the polls in order to cast a ballot in another person’s name.
You can find anecdotal evidence for each of these. First, you need to separate fraudulent intent from basic ignorance. The latter has been found much more often than the former. Ignorance of voting laws has a much higher probability than criminal conspiracy, and ignorance is usually unbiased for either side. Also, state laws regarding the reinstatement of voter privileges to felons who have served their sentences are varying and confusing, without even getting into the “fairness” of such rules. And there are more citizens with felony convictions than you probably think. By one estimate, 8.6% of the adult population in the U.S. has a felony conviction, including up to 35% of black adults in some states. 
Peoples’ confusion in their own citizenship status is also far more widespread than commonly thought. Millions of Americans, many elderly, would have a difficult time “proving” their citizenship on short notice if pressed, due to poor or lost documentation. Most Americans do not have passports, and a driver’s license does not prove citizenship. We just don’t typically challenge this for “normal people” who look like us. Unfortunately, people have been prosecuted for “ignorant voting,” but ignorance is not fraud, nor is it conspiracy.
And so, some numbers. In the Third Congressional District of Kansas, the most competitive in the state, over 343,000 people voted in the 2016 election. No audit that I have ever seen has found more than five ignorant/intentional illegal voters in any single district election. But even if there were 10, this would mean that the probability of an illegal vote would 0.003%, which is 3 thousandths of a percent, or 3 in 100,000. Other audits have found this number maxing out, including (and mostly) the “ignorantly illegal” votes, at one in 10,000. This level of error has only a tiny effect on election outcomes. 
What level of fraud would it take to “steal” and election?
Clearly, a voter fraud rate even of one-hundredth of one percent (34 people in the Third District) is very rarely enough swing an election, and in those rare cases, the numerous other causes of miscounts will far exceed this rate.  Then you have to look at the probability that all voter fraud is heavily biased in one political direction, which it is not. In other words, fraudulent voters are most likely to cancel one other out. 
If there is a grand conspiracy of thousands (or millions, as Donald Trump has claimed) of illegal votes, then one proven technique can determine cheating at that required high level. Auditing of votes cast in critical elections matched to voters by basic statistical sampling would be both inexpensive and mathematically reliable. But most people don’t understand statistical sampling beyond the squishy “quasi-math” of political polls.  However, no professionally-done audits of important national elections have found results-changing problems due to voter fraud in recent years. 
What is the probability that tight voter ID and registration laws disenfranchise legal voters?
High. Really high. The primary proposed political remedies to this mostly-imagined voter fraud are twofold. First, require specified forms of information at the polling place. Second, demand greater proof of citizenship and other qualifications at the time of voter registration.
The problem with every one of these initiatives is that the probability of ensnaring in the net thousands of citizens who have every legal right to vote is very high. To start with, the legislators often embed political criteria on various forms of identification, ensuring that many college students, elderly, and other people of less-than-permanent domicile are denied their right to vote. In Texas, a gun permit is an official ID for voting purposes, but a Texas university ID is not. This is an intentional biasing of eligible voters.
Kris Kobach has been continually hammered by the courts due to his attempts, as Secretary of State of Kansas, to force higher documentation requirements for registering to vote than federal legislation demands, and has been held in contempt of court.  And lately, even citizens legally holding valid U.S. passports are being detained at the border and challenged. Intimidation of minorities is a very real, well-documented tradition in ballot places and other public places across the United States.
Add to this documented incidents of aggressive purging of voter rolls, and it is easy to quantify thousands of legal voters being disenfranchised in these states, which means that the probability of legal voters kept from voting in these states likely exceeds the number of non-legal voters by several “orders of magnitude” (powers of 10). In short, only one of these is a significant problem of mathematical significance. Guess which one.
Kris Kobach knows the math
I am assuming that Kris Kobach is an intelligent guy, and “knows the math” both of fraudulent voting (a tiny probability) and voter disenfranchisement (a very high probability). Let’s be frank. His motivation is political and nativist. He does not believe in the right of all citizens to vote, as many of them are very likely not to vote his way.
Yet Kobach frames his argument in ways targeted at the vast majority of his constituents who do not know the math. That is both disingenuous and anti-democratic.
- Rice University. “Hand counts of votes may cause errors.” ScienceDaily, 2 February 2012.
- Suede, Michael. “What Percentage of The US Adult Population Has a Felony Conviction?” Libertarian News, 5 June 2014.
- Nineteen foreign nationals are being prosecuted for voting in North Carolina’s 2016 election out of 4.8 million voted cast (0.0004%), but fraudulent intent has not yet been proven. Wines, Michael. “19 Noncitizens Voted Illegally in 2016 in North Carolina, U.S. Charges.” The New York Times, 25 Aug. 2018.
- See this earlier post about an important Virginia state election that came down to a tie, and the drawing of lots to determine a winner.
- One of the few identified cases of “voter ID” fraud in 2016 was a Trump voter, and this case highlighted how most voter impersonation fraud is not due to conspiracy, but is rather just plain stupidity at work. Gordon, Michael. “DA Won’t Prosecute Woman Who Followed Mom’s ‘Dying Wish’ and Cast Illegal Vote for Trump.” Charlotte Observer, 26 April, 2017.
- Political polls usually use unreliable and biased methods (e.g., calling landlines) to ask questions of anonymous people, who may or not be forthright, regarding a future intention, and then try to adjust for known biases in the sample. On the other hand, an after-the-fact statistical sample of votes cast determines that a purported voter either was or was not legally registered, and indeed was the person who voted. The latter is of much higher mathematical reliability in finding fraud or errors.
- The 2000 Bush-Gore recount in Florida was not about voter fraud. Instead, it revealed the more basic, and much higher-probability human-error causes of miscounts, as well as the ability of supposedly apolitical courts to put their thumb on the scale.
- Chappell, Bill. “Judge Tosses Kansas’ Proof-Of-Citizenship Voter Law And Rebukes Sec. Of State Kobach.” NPR, 19 June 2018.