Vote Rigging

Launch worksheet

Launch solution 

The Scenario

You work for the federal intelligence service and have been commissioned to investigate suspicions that the recent presidential elections were rigged using voter fraud. Using mathematical analysis, you’re going to investigate several promising leads that will help tell you whether your country is under the control of an illegitimate leader! Because the stakes are so high, you’re going to need to conduct this investigation covertly, using whatever information you can get your hands on.

Math Scenario 1: Vote Counting

The picture below shows the various stages of the election process:

After business hours have finished, you let yourself into one of the contentious voting stations and examine their records.

At this station, each tally was counted by two different people, who had to get the same total. If they didn’t get the same total, then there was a further recount.

You find that of the 10 counts done at that polling station, 8 required further recounts. In each case, the highest total was eventually accepted as being the correct one.

If people were equally likely to under count as to over count, we can calculate the normal chance that all 8 errors were due to under counting:

Chance of all errors being due to under counting = (chance of one undercount)^number of counts

Chance of all errors being due to under counting = 0.5^8

Chance of all errors being due to under counting = 0.0039

All other things being equal, that’s a 0.3% chance of that result happening, which is suspicious but not enough evidence by itself. You continue the search…

Math Scenario 2: Vote Buying

Using your secret intelligence network, you gain access to the bank records of the 35000 voters in a particular electoral area. 

Normally in a non-election year, the average person’s bank account in that area has an average of $850 deposited into it over the course of a month.

In the month leading up to the election, the average account in the area had an average of $4350 deposited into it.

We can calculate the factor by which that amount is above the normal monthly deposit rate:

Deposit ratio = election month deposit /  normal monthly deposit

Deposit ratio = 4350 / 850

Deposit ratio = 5.118

We can then calculate by what percentage deposits increased over normal in the month preceding the election:

Increase factor = increase / normal amount

Increase factor = (election deposit amount – normal amount) / normal amount

Increase factor = (4350 – 850) / 850

Increase factor = 4.118

That’s an increase of more than 400% in deposits in the month preceding the election! 

Math Scenario 3: Polling


Vote polling isn’t always reliable, but it can sometimes be used as an indicator of whether an election result is reasonable or not.

During the last electoral debate, an extensive poll across all demographics across the entire country resulted in this:

  • Candidate A: 54%
  • Candidate B: 42%
  • Candidate C: 2.5%
  • Candidate D: 1.5%

After the election, these were the results:

  • Candidate A: 47%
  • Candidate B: 49%
  • Candidate C: 3%
  • Candidate D: 1%

We can calculate the swing against each voter, both as an absolute percentage change and a relative increase or decrease in proportion of the votes:

Absolute Percentage Changes

  • Candidate A = 47 – 54 -7%
  • Candidate B = 49 – 42 = 7%
  • Candidate C = 3 – 2.5 = 0.5%
  • Candidate D= 1 – 1.5 = -0.5%

Relative Increase or Decrease in Proportion of Vote

  • Candidate A: 47 / 54 = 0.8704
  • Candidate B: 49 / 42 = 1.167
  • Candidate C: 3 / 2.5 = 1.2
  • Candidate D: 1 / 1.5 = 0.6667

All candidates appear to have had large swings for or against them. In the case of Candidates A and B, the swing changed the outcome of the election.

Real Life Example: 2016 U.S. Election

After being elected president of the United States of America in 2016, Donald Trump claimed voter fraud resulted in him not winning the popular vote, mentioning that many millions of illegal votes were cast.

“You have people that are registered who are dead, who are illegals, who are in two states. You have people registered in two states. They’re registered in a New York and a New Jersey. They vote twice. There are millions of votes, in my opinion.”

One of the states of contention was New Hampshire, with a population of about 1.3 million as of 2016.

Donald Trump got 46.5% of the vote, while Hilary Clinton got 46.8% of the vote.

Let’s say voter fraud was committed by continuing to vote using dead people. How far back would you have to go?

According to this site, the death rate per 100000 people per year in New Hampshire is 706.2 people, as of 2014.

So that means:

Deaths per year = death rate per 100000 people * population / 100000

Deaths per year = 706.2 * 1300000 / 100000

Deaths per year = 9180

We can calculate that as a percentage of the state population (using either that figure or the original rate figure):

Deaths per year = 706.2 / 100000

Deaths per year = 0.007062

Which is 0.7% of the population. To change the vote to a draw, Trump would have needed Hilary to win (46.8 – 46.5) = 0.3% less of the vote.

So if you could vote for Hilary using all the people who have died, we can work out how many months or years of dead people we’d need to change the result:

Required dead people history = required percentage / percentage of dead people per year

Required dead people history = 0.3 / 0.7

Required dead people history = 0.4286 years

Required dead people history = 5.14 months

However, to pull of the fraud successfully, without detection, you could probably only do this with a very small percentage of dead people. Let’s say you could only use 5% of dead people as illegitimate voters.

This changes how far back you’d need to go:

Required dead people history = 0.4286 years* 1 / 0.05

Required dead people history = 8.572 years

While this claim will be either verified or debunked in due course, there are many other ways to commit electoral fraud including intimidation, vote buying, ballot stuffing, vote ballot destruction… you can read the many ways it occurs here.