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Spotting Random Fraud

Let’s say we have an election and 20 precincts report their results. Here’s the total number of votes cast in each precinct:

3271                            2987                2769                            3389

2587                            3266                4022                            4231

3779                            3378                4388                            5327

2964                            2864                2676                            3653

3453                            4156                3668                            4218

Why would you suspect fraud?

Before you answer that, let me ask you another question. Would you please write down a random number between one and 20?

Asking you to write down a random number seems like an innocent request. But the word “random” invokes some unusual behavior. It turns out that we all have in our minds a definition of “random” that’s not quite … well, random. Does the number 17 seem random to you? Most people would say, “Sure. That’s pretty random.” Do the numbers 10 and 15 seem random to you? Most people would say, “No. Those aren’t random numbers.”

Why do we have a bias against 10 and 15? Why do we say they aren’t random? Probably because we often round our numbers so that they end in zeros or fives. We say, “I’ll see you in five minutes (or 10 minutes or 15 minutes)”. We rarely say, “I’ll see you in 17 minutes”. In casual conversation, we use numbers that end in zeros or fives far more often than we use numbers that end in other digits. Because we use them frequently, they seem familiar, not random.

So, if we want numbers to look random – as we might in a fraud – we’ll create numbers that fit our assumptions of what random numbers look like. We’ll under-represent numbers that end in fives and zeros and over-represent numbers that end in sevens or threes or nines. But if the numbers are truly random, then all the digits zero through nine should be equally represented.

Now look again at the reported numbers from the precincts. What’s odd is what’s missing. None of the twenty numbers end in five or zero. But if the numbers were truly random, we would expect – in a list of 20 — at least two numbers to end in zero and two more to end in five. The precinct numbers are suspicious. Somebody was trying to make the numbers look random but tripped over their own assumptions about what random numbers look like.

Moral of the story? If you’re going to cheat, check your assumptions at the door.

By the way, I ask my students to write down a random number between one and 20. The most frequent number is 17, followed by 3, 13, 7, and 9. There is a strong bias towards odd numbers and whole numbers. No one has ever written down a number with a fraction.

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