Many election jurisdictions in the United States need to randomly select precincts, or batches of ballots, to audit after an election. This article provides step-by-step instructions for how to do this correctly using dice. It also explains some of the advantages of using dice over other methods.

This article is intended for election officials, as well as members of the public interested in improving election auditing procedures in their area.

There are a number of correct ways to use dice to randomly select precincts to audit (as well as many incorrect ways, if one isn't careful). The method described here is the one described in a 2006 paper by Arel Cordero, David Wagner, and David Dill (see [CWD] referenced at the bottom). We chose this method both for its simplicity and relative efficiency. Also, we simplified the description so that, for example, familiarity with math notation isn't needed. (We do use letters to stand for numbers, though.)

## Instructions

Even though the steps below can be used without change to select not just precincts but also batches of ballots, for simplicity we always say "precincts."

### 1. Preparation

Before doing the selection, you need to determine two key numbers: the number of dice to roll each time, which we call \(R\) (for "roll"), and a "dividing factor" (also a whole number), which we call \(D\) (for "dividing"). These numbers depend on the total number of precincts you have. You will need a calculator.

To determine \(R\) (aka the number of dice to roll each time), take the total number of precincts you have, subtract 1, and then note the number of digits that number has. This number of digits is \(R\). For example—

- For 11 to 100 precincts, \(R\) is 2.
- For 101 to 1,000 precincts, \(R\) is 3.
- For 1,001 to 10,000 precincts, \(R\) is 4, and so on.

To determine \(D\) (aka the dividing factor), take the number 1 with \(R\) zeros after it (e.g. 1,000 if \(R\) is 3). Then, use a calculator to divide that number by the total number of precincts you have. Finally, round the answer *down* (i.e. by removing the decimal part, if any). For example, if the total number of precincts is 170, then \(R\) would be 3. Divide 1,000 by 170 to get 5.8823.... Then round that down to get 5 for \(D\). As another example, if the total number of precincts is 200, then \(R\) would again be 3. Dividing 1,000 by 200 gives 5 exactly. Since 5 has no decimal part, rounding down keeps it at 5. So \(D\) would still be 5.

With the preliminary calculations done, now make sure you have \(R\) different ten-sided dice. It's a good idea if each one is a different color so they can be rolled together. For example, if \(R\) is three, you can purchase three ten-sided dice in the colors red, white, and blue. (Ten-sided dice are sold in many different colors.) You can see a picture of what they look like on the right.

TODO: finish the rest.

## References

- [CWD] Arel Cordero, David Wagner, and David Dill, "The Role of Dice in Election Audits - Extended Abstract," June 16, 2006, 8 pages, preprint.