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March 29, 2018

RPG Math: Calculating the Average Dice Roll

RPG Math

Welcome to the first post of RPG Math. This is a series of posts that will dissect mathematical formulas and equations used in various RPGs and teach you how to use them to your advantage. We are going to begin this series with an extremely basic, but constantly utilized formula. The average dice roll calculation.

When you roll a die or group of dice, let’s say 2 6-sided dice, they will total any number between 2 and 12. Probability dictates that there will be a certain rate in which these dice will roll a certain value. I’ve written a post on basic probability before should that interest you. What this post will be explaining is how to calculate what we can expect the sum of the dice to be once they are rolled.

For this equation all you will need to know is some basic addition and multiplication, or simply how to use a calculator.

Definitions

Dice Naming Conventions (xdy)

Throughout this post I will be using shorthand to talk about different combinations of dice rolls. To make this easier I will be calling them xdy. The X in this case is the quantity of dice rolled. The Y in this case is the number of sides on the dice. For example 1d6 is a 1 six-sided die.

Random Number Generator (RNG)

This is something that will select a number between a pre-determined set of constraints. Most importantly, this generator does not favor any outcome more than another possible outcome. Dice are an example of a random number generator. When rolled, 1d6 will randomly “select” any number between 1 and 6 (unless it’s weighted). Here’s a post I wrote that goes into more detail on RNG.

Constraints

This is a set of numbers that the random number generator has to stay inside of. For example our 1d6 has to be within the numbers 1 and 6. It cannot select a 0 or a 7+ as they are outside of its constraints.

Average

The typical value of an outcome. This value is in the middle of your random number generator and therefore what you can mathematically expect to be selected.

The Average Dice Roll Formula

( ( Max Die Roll + 1 ) / 2 ) * Number of Same-Sided Dice

That’s all you have to do. It’s just some basic addition wrapped around division and possibly multiplication if you are rolling many of the same-sided dice. Remember to always use PEMDAS! These are the exact steps to solve an equation using the Average Dice Roll formula:

1) Input your values for the Max Die Roll and the Number of Same-Sided Dice.
2) Add 1 to your Max Die Roll.
3) Divide Max Die Roll + 1 by 2.
Optional: 4) Multiply the value of step 3 by the number of dice that have the same number of sides.
Optional: 5) Repeat for other sets of unique dice and add the sums of all relevant dice together.

The big question is probably: “Why are we using the Max Die Roll +1 and not simply the Max Die roll?”. Great question. With any die we use their constraints will start at 1 and not 0, therefore we need to compensate for this in our formula by adding 1.

Another option for finding the average dice roll is to add all of the possible outcomes together then divide by the number of sides the die has. For example, to find the average dice roll of 1d4 you would add 1, 2, 3, and 4 together and divide by 4. You would come out with 2.5. This is much more tedious than our formula though!

Examples

1d20

Let’s start with a basic one. If you are into D&D or other d20 based games this will be a great one to know.

1) Average = ( ( Max Die Roll + 1 ) / 2 ) * Number of Same-sided Dice
2) Average = ( ( 20 + 1) / 2 ) * 1
3) Average = ( 21 / 2) * 1
4 ) Average = 10.5 * 1 = 10.5

You can now find the average outcome of when you make a weapon attack or use a skill. All you have to do now is add your Proficiency Bonus and the relevant Ability Score Bonus to 10.5. It’s as easy as that!

2d6

We are rolling 2d6 which is a common roll in many games such as Apocalypse World which runs entirely off of 2d6 rolls.

1) Average = ( ( Max Die Roll + 1 ) / 2 ) * Number of Same-sided Dice
2) Average = ( ( 6 + 1) / 2 ) * 2
3) Average = ( 7 / 2) * 2
4 ) Average = 3.5 * 2 = 7

The average dice roll of 2d6 will be 7. This becomes faster as you memorize the average dice roll of a singular die. If you know that the average roll of 1d6 is 3.5 you’ll just skip to multiplying your average by the number of dice you’re rolling in the future.

1d8 + 3d6

Let’s do a more complicated one now. This equation is for the average dice roll of a D&D 5e Rogue’s sneak attack with a Rapier. This uses 2 unique dice rolls, but we are summing them together to find their average damage as it is all a singular attack.

1) Average = ( ( ( 8 + 1 ) / 2 ) * 1) + ( ( ( 6 + 1) / 2 ) * 3 )
2) Average = ( ( 9 ) / 2) * 1 ) + ( ( 7 ) / 2 ) * 3 )
3) Average = ( 4.5 * 1 ) + ( 3.5 * 3 )
4) Average = 4.5 + 10.5 = 15

The average dice roll of 1d8 + 3d6 is 15. This can be done with any combination of dice imaginable.

Use Cases

The average dice roll formula is useful alone, but it is also necessary for estimating the outcome many mechanics in RPGs. For example, I used it to create my Great Weapon Master and Sharpshooter Calculator.

You can also use it for many other D&D 5e mechanics estimators such as calculating the chance to hit an enemy or make a saving throw. These will absolutely be the subject of their own post in this series in the future.

This formula is not constrained to just physical dice. Any instance when you know the constraints of a random number generator you can use this formula to predict the average value of it. For example, the drop % of a particular piece of loot in an MMO would include this formula.

Conclusions

Knowing how to find the average dice roll can give you an advantage when you have to make a risky decision. Is it worth it to climb that mountain if the DC is 15? Well since we know the average dice roll of 1d20 is 10.5 and you have 18 STR and +3 proficiency you have an average dice roll of 17.5 with your athletics check. Sounds like it’s well worth the risk!

As with any prediction or estimation that uses a random number generator, this is not going to guarantee an outcome. Calculating the average dice roll will simply give you more information on how likely you are to roll a certain number. Ultimately you will have to accept that you always have a chance to fail despite having calculated a high average.

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