This article provides step-by-step instructions for how to use dice properly to randomly select precincts, or batches of ballots, after an election. Many election jurisdictions in the United States need to do this after an election for auditing purposes. This article 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 of using dice to randomly select precincts (as well as many incorrect ways, if one isn't careful). The method described in this article is the one described in a 2006 paper by Arel Cordero, David Wagner, and David Dill (also see [CWD] referenced at the bottom). We chose to highlight this method both for its simplicity and efficiency over other methods. 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.) Lastly, the method we describe here is *not* suitable for risk-limiting audits. For those a different procedure is needed.

## Advantages

Rolling dice is a good method for doing a random selection because it is simple and straightforward, easy to observe, and is observably random. Confidence can also be increased by letting members of the public participate by doing some of the dice rolls.

Other processes like drawing slips of paper from a hat or barrel don't have these same advantages. For example, if drawing slips of paper, it's hard to know if the slips of paper are evenly mixed, and there is no clear way for the public to confirm this. Also, if you're choosing from among hundreds or dozens of precincts, there is no easy way for the public to know for sure that every precinct is present in the hat. Finally, it's also hard for members of the public to be sure that sleight-of-hand techniques aren't being used when choosing the pieces of paper.

## Terminology

This section contains definitions of terms that we don't define elsewhere in this document.

Some jurisdictions need to pick randomly from multiple groups of batches separately. For example, a jurisdiction might have in-person ballots grouped by precinct, vote-by-mail ballots grouped by batches, and provisional ballots grouped in a different set of batches. If the jurisdiction has to pick randomly from each of these groups separately, we call them **pools**. Thus, in this example there would be three pools that each require a random selection process: one for precincts, one for batches of vote-by-mail ballots, and one for batches of provisional ballots. Whether the pools do in fact need to be treated separately may be up to state or local statute, or it could be left as an administrative decision.

## Instructions

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

### Advance Preparation

We start by describing the steps you can do in advance.

#### Step 1: Purchase enough dice

You will need to purchase a set of dice in advance. You will need *ten-sided* dice rather than standard six-sided dice. You can see a picture of what ten-sided dice look like on the right. You should be able to find ten-sided dice for sale at a local games shop. Otherwise, they are widely available online for purchase.

Since it may take time to purchase dice (e.g. if you're ordering them online), you should acquire them far enough in advance and make sure you will have enough. The number of dice you need in your set depends on the number of precincts or batches you will have. If you don't know in advance how many precincts or batches you will have (especially in the case of batches), you should estimate the absolute maximum number you might wind up having. It's okay to overestimate here. Also, when coming up with your estimate, you only need to think about your largest pool of batches rather than combining them together. For example, if you will have two pools of batches, one for precincts and one for vote-by-mail ballots, you only need to estimate for the pool you think will have more.

If you will have no more than 100 precincts or batches in your largest pool, you will need two dice in your set. If you will have no more than 1,000, you will need three. For 10,000, you will need four. And so on. (This is the same as the number \(R\) we will be discussing below.)

When purchasing dice, it's a good idea if each die in the set is a different color. This way they can be rolled together all at once and you can still tell them apart. For example, if you need three dice, you can purchase three ten-sided dice in the colors red, white, and blue. (Ten-sided dice are sold in a variety of different colors.)

Finally, it's a good idea to purchase extras in case you lose any, or in case you underestimated the number of batches. For example, you can purchase one or more extra sets, and you can make sure each set has one more die than you estimated you would need (also of a different color from the other dice in the set).

#### Step 2: Choose the precinct lookup sheet format

To do the random selection, you will need a document we call the "precinct lookup sheet" (or "batch lookup sheet," in the case of batches). This is a document or spreadsheet that lets you look up the correct precinct each time you roll the dice. Both participants and observers will refer to this document during the selection.

There are two styles of precinct lookup sheet: the two-column format and the four-column format. You will need to decide which style to use before completing the next steps. The two-column format is slightly easier to prepare and makes for a more compact lookup sheet. The four-column format is less compact, but it should make the selection process easier to conduct and follow.

It's best to read the rest of this document before deciding which format to use. If you're still not sure or if this is the first time you'll be using dice for a random selection, the four-column format is a good choice. Later sections of this document describe the lookup sheet in more detail and how to make it. You can only make a lookup sheet after you know your complete list of precincts or batches.

#### Step 3: Prepare the selection worksheet

This section describes how to create what we call the "selection worksheet," which is different from the lookup sheet. Whereas the lookup sheet is for reading information from, the selection worksheet is for filling out. Also, unlike the lookup sheet, you *can* prepare the selection worksheet before you know the full list of precincts or batches. This is because the blank worksheet doesn't include any information specific to the election.

The selection worksheet is a document or spreadsheet to which the dice rolls and corresponding precincts are recorded after each roll of the dice. The worksheet can be used to provide a written record of what happens during the selection. Also, if you're using the two-column format of the lookup sheet, the selection worksheet has an extra column to help you determine the precinct corresponding to a dice roll. (This extra column isn't needed for the four-column format.) The instructions for filling out the worksheet will come later in this document.

We first review the selection worksheet to make if you're using the four-column format of the precinct lookup sheet. This version of the worksheet has two row entries to fill out at the top, and four columns of information to fill out at the bottom (one row for each roll of the dice). A picture of what to prepare appears below.

We next review the selection worksheet to make if you're using the two-column format of the precinct lookup sheet. This worksheet has three row entries to fill out at the top, and five columns of information to fill out at the bottom (one row for each roll of the dice). A picture of what to prepare appears below.

If you're using the two-column format, and if the selection worksheet will be filled out while using a computer spreadsheet instead of on paper, then the "Roll ÷ S" and "Index" columns can be set up in your spreadsheet program to automatically be populated when the number in the "Roll" column is entered. This isn't strictly necessary, but it's an available option and could make things easier.

#### Step 4: Determine the helper numbers

Before doing the selection, you need to determine two key "helper" numbers for each pool. You can do this step for a pool once you know the exact number of precincts or batches in that pool.

The two numbers you need to determine are— (1) the number of dice to roll each time, which we call \(R\), for "roll," and (2) a "scale factor" (also a whole number), which we call \(S\), for "scale." These numbers depend on the total number of precincts you have.

To determine \(R\), the number of dice to roll, 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 \(S\), the scale factor, take the number 1 with \(R\) zeros after it (e.g. 1,000 if \(R\) is 3). Then, use a calculator and 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 \(S\). 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 \(S\) would still be 5.

#### Step 5: Make precinct lookup sheet

Next, you need to make the precinct lookup sheet we first mentioned in Step 2, one for each pool. As with Step 4, you can do this step for a pool once you know the exact number of precincts or batches in the pool.

Recalling from Step 2, the lookup sheet is a document or spreadsheet that lets you look up the right precinct or batch each time the dice are rolled. Both election staff and observers will be referring to this document during the selection, so you should be prepared to distribute a copy.

In its simplest form, the lookup sheet is simply a numbered list of the names of all precincts. In Step 2, you chose whether to use the 2-column or 4-column format of this sheet.

If you chose to use the 2-column format, create a numbered list of the precincts as pictured below. The numbers should start at 0, and the numbers should go up by one with each precinct. No numbers should be skipped. We call each number the "index" of that precinct.

If you chose the 4-column format, start by following the instructions for the 2-column format. Then, add two more columns to the grid called "Min Roll" and "Max Roll," as pictured below.

To fill the "Min Roll" column for a certain row, multiply the index number for that precinct by the number \(S\) you determined in Step 4. To fill the "Max Roll" column, add \(S - 1\) to the "Min Roll" value. In other words, "Max Roll" is one less than "Min Roll" of the next row. The Min Roll and Max Roll numbers will let you and any election observers see immediately what precinct corresponds to a roll of the dice, without having to do any additional arithmetic. If you're using a spreadsheet program, you should be able to create formulas for the Min Roll and Max Roll columns so these two columns can be filled for you automatically.

Finally, for convenience, you can add a row at the end that says "ROLL AGAIN" for any higher numbers, as in the picture.

#### Step 6: Complete the pre-selection checklist

Here is a checklist of some additional things that should be completed before doing the actual random selection:

- Make sure you are done counting all of the ballots and votes.
- Make the final vote totals for all of the precincts and batches available to the public, for example by posting them online. This way everyone can know what vote totals you are checking.
- Announce to the public when you will be conducting the random selection.

### Actual Selection

#### Step 7: Conduct the selection

When doing the random selection, it is best to have a document ready that explains the procedures you will be following. The document should cover things like:

- The order in which the digits of the number will be read off the dice, using the colors. For example, if there are three dice with colors red, white, and blue, it might be: "first red, then white, then blue."
- What happens if a die falls on the floor? Do you roll that die again, or do you use whatever number is shown on the floor?

You should also have ready to distribute a copy of the precinct lookup sheet you will be using.

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.