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In the realm of tabletop role-playing games (TTRPGs), dice and dice probabilities are more than just tools of chance—they are the enchanted artifacts that shape the fate of heroes, villains, and the world.

For a game designer, understanding the basic statistics behind multisided dice is akin to mastering a magical art. It is the key to crafting balanced encounters, designing fair challenges, and ensuring that every roll of the die brings excitement and suspense to the table.

Knowing the probabilities behind dice rolls helps create an engaging gameplay experience. With this information, you possess a deeper understanding of what the players (not the characters) actually do.

How much time will the players spend in combat? How many times do you actually have to roll a specific die in order to overcome challenges? How do health potions or magic weapons alter playability? With a solid understanding of basic statistics, TTRPG designers can answer these questions.

The Importance of Dice Probability

Why Do Statistics Matter in TTRPG Design?

It’s important to understand certain basic fundamental statistical terms and concepts when designing a tabletop RPG. You need to understand what your players will actually be doing at the table. They are not going on epic adventures; they are rolling dice.

While the narrative of a TTRPG may whisk players away on grand quests and heroic journeys, the reality at the table is grounded in the mechanics of dice rolling. Every decision, every action, and every outcome in the game is mediated by the roll of the dice.

As a game designer, grasping the basic statistical principles behind these rolls is crucial. The players’ experiences are fundamentally shaped by these numbers, not just by the story you tell. To design a game that truly engages and challenges your players, you must first understand what those dice are doing and how often certain outcomes will occur.

How Do Statistics Transmute into Fun?

System mechanics and dice rolls simulate actions and challenges in the real world. To create an immersive and fun game world, you need to understand and manipulate the probability of these “actions.”

The challenge for a game designer is to simulate real-world actions, like swinging a sword or picking a lock, in a way that feels both fair and exciting. By understanding the dice probabilities behind each roll, you can craft mechanics that accurately represent the difficulty of these tasks. This allows you to create a world that feels coherent and immersive, where the outcomes of actions are both believable and engaging.

Ask the OG Game Designing Professionals!

Long before Gary Gygax and Dave Arneson wrote their first d20 adventure, Las Vegas made an industry of understanding dice probabilities. Everything in the casino is designed to keep patrons in their seats and enjoying themselves. How often a game pays out is precisely and statistically balanced to push the brain’s pleasure center while maintaining the money flowing into the casino’s coffers.

Much like a Vegas Baron, a TTRPG designer strives for a finely tuned balance between risk and reward. Game writers must search for an equation that ensures players win often enough to stay excited without losing a sense of danger.

By adjusting the difficulty of tasks and the rewards for success, you can control the pacing and tension of your game, making sure players remain engaged and eager to roll the dice again.

Dice Probabilities Explained

Understanding how dice probabilities work is crucial for any TTRPG designer. Before exploring how dice probabilities shape your Tabletop RPG, it’s essential to grasp some fundamental statistical concepts. These foundational terms will help you formulate engaging and exciting game mechanics.

Single-Outcome Probability of Dice Rolls

(Or Exact Outcome Probability)

When you roll a die, each face has an equal chance of landing face up. The term Single-Outcome Probability refers to the likelihood of rolling a specific number on that die. This concept is straightforward but crucial.

You gotta understand this before moving on to concepts that provide you with more control over gameplay.

Each number on a d20 possesses the same potential for appearing on a random dice roll. Whether it’s a devastating 1, a glorious 20, or a mediocre 10, you have a 1 in 20 shot of hitting that number when tossing the dice — or a 5% probability of landing on any specific number.

To figure that out, you simply need to calculate the percentage of 1 in 20. Remember, you are looking for the probability of one single number. It doesn’t matter if that number is 1, 10, or 20. It is just 1 number out of 20, so you divide 1 by 20

1 ÷ 20 = .05

… or 5%

Cumulative Probability of Dice Rolls

This is where dice statistics become fun. Many game mechanics hinge on the likelihood of rolling above a target number, a threshold, or a score. Understanding how this threshold works provides the game designer with the tools needed for shaping gameplay. Cumulative probability is that tool.

Unlike Single Outcome Probability, which focuses on the likelihood of a specific number resulting from a roll, Cumulative Probability considers all possible outcomes that meet or exceed the desired threshold. It accounts for all the potential outcomes that result in a success.

For example, if a player needed to roll and 8 or higher to overcome a challenge or attack an enemy, the probability includes all numbers from 8 to 20. Rolling a 9, 16, 12, or 20 will get the job done.

Instead of calculating the chance of a single number turning up, you look at the total number of successful outcomes. There are 13 numbers from 8 to 20 that will fit the bill, overcome the challenge, or hack off the orc’s head. Therefore, you have a 13 in 20 chance of success.

To find this percentage of success, you divide that cumulative number of successful outcomes by 20.

13 ÷ 20 = .65

…or 65%

Using Statistics to Design TTRPGs

Utilizing Cumulative Probability is at the core of tabletop role-playing game design. It is the fundamental tool that underlies balance, engagement, and consistency, but is often overlooked. Most games hide this essential tool behind a pile of modifiers.

Modifiers: The Hidden Math Behind RPG Design.

Modifiers are a core component of many TTRPG systems. They provide players with tools to manipulate the game and interact with the in-game world. However, players might be surprised to learn that modifiers often act as a smokescreen, obscuring the real math behind RPG mechanics.

When a game designer chooses a threshold for success, they decide the number of rolls players make before the orc’s head is lopped off or the locked door is picked. If a designer wants to maintain engagement with a 65% success rate, they set the threshold at 8. But the player doesn’t know that; they can’t know that. If the threshold remained consistent and relatively low, players might get bored.

To maintain engagement and the dice probabilities a secret, TTRPG mechanics utilize modifiers to mask the statistics. These modifiers not only increase the average score rolled by a player but also compel the designer to scale up the threshold for success. Since modifiers typically increase as characters level up, thresholds must also scale up at an equal rate.

For example, if the RPG designer sets the threshold so players have a 65% chance of success, then the number needed to be rolled is 8. If the player has a +5 modifier, then to maintain the 65% threshold, the game designer simply adds +5 to the threshold.

8 + 5 = 13.

Leveling Up: The Illusion of Scaling Power

One of the key challenges in TTRPG design is maintaining balance as players level up and become more powerful. As players level up, they often gain higher modifiers to their rolls, making them more likely to succeed on any given check. To preserve the game’s challenge and excitement, game devs raise the visible thresholds at the same rate as the modifiers.

When a character levels up and sports a shiny new +8 modifier, the challenges they face scale to match. To maintain that same 65% chance of success, the threshold bumps up to 16. The player may feel stronger, faster, and cooler, but they still have the same 65% chance of success. This approach keeps the game balanced while providing the illusion of scaling power.

The Average Roll

Understanding the average roll is just as important as the. While single-outcome and cumulative probabilities help determine the likelihood of success for specific actions, the average roll provides insight into the overall pace and balance of your game. This concept becomes especially important when designing mechanics that involve the accumulation or depletion of resources over time, such as health points (HP) and damage.

The Role of Average Rolls in Game Balance

Whenever dice probabilities are used to determine accumulation or depletion of a resource, the average score of that dice must be examined. The depletion of HP, fuel, or a magical power (manna) are examples of a resource that may be lost. The accumulation of damage, distance traveled, or speed are common examples of resources gained.

What really matters is how long it takes a player to acquire or deplete these resources. If a player’s character loses their health points too quickly, the game is over and everyone goes home pissed off. If the enemies battleship accumulates damage to quickly, then the players don’t find the encounter challenging and get bored.

Essentially, the average role of the dice comes into play most frequently when determining the pace of a game. The accumulation and depletion must occur at a rate that is entertaining and balanced. No game designer can guarantee an exact number of rounds in which an enemy will go down. However, effective tabletop RPG mechanics focus on the average dice roll when accumulating or depleting these types of resources.

Calculating the Average Dice Roll

Finding the average role of a die is easier than convincing your players to stop meta-gaming. In fact, there are a few ways you can go about finding the average of any given dice. Whereas the only certain way to keep your players for meta-gaming is to throw a all-powerful red Dragon at them and giggle maniacally as they cry.

In other words, math is the easy part of writing an RPG.

Let’s start crunching numbers. To find the average of any dice, take the lowest number on the die and add it to the highest number. Then divide the sum in half. Why don’t we start with the standard six-sided die (d6)?

The lowest number is 1, the highest number is (of course) 6, add them together and you get lucky number 7. Divide seven and half, and you end up with the average score for a d6, 3.5.

d6 average roll

step one

1 + 6 = 7

Step two

7 ÷ 2 = 3.5

The same formula applies to any other die. Got a d20? The lowest number is 1, the highest is 20. Add them to get 21, then divide by two, and you get 10.5. This is the average roll for a d20.

d20 average role

Step one

1 + 20 = 21

Step two

21 ÷ 2 = 10.5

To make it easy on you, I went ahead and wrote up the averages for the most common die found in TTRPGs.

Dice Type	Average Roll
d4	2.5
d6	3.5
d8	4.5
d10	5.5
d12	6.5
d20	10.5

Using Statistics to Build TTRPG Mechanics

Alright, it’s time to put together what you’ve learned. Let’s say we’re designing a system where players have a 65% success rate. This means that when they start rolling their d20s, they should succeed about 2 out of 3 times.

For now, let’s drop modifiers from the discussion. We’ve already established that they’re something of a smokescreen, so we’ll avoid convoluting our math for the moment. We can always add modifiers to tweak the roll and the threshold later.

Let’s assume we are rolling a resource metric. Health points (HP) is the obvious choice for this example, so let’s roll with that… 😊

Let’s say the damage die is a d6. For every successful d20 role, you’re doing a d6 average’s worth of damage — which is, of course, 3.5.

Therefore, at the end of three turns, you have probably landed 2 hits and dealt an average total of 7 points of damage. Initially, this may not seem like a lot, but remember, most TTRPGs have multiple players. If three players dish out roughly 7 points of damage with each successful attack, then after everyone has had three turns, the enemy has taken an average of 21 points of damage.

Now You Know!

This is how you predict the flow of combat. This is how you determine the pace of a game. If the enemy has 21 health points, they’re probably toast once everyone has taken three turns. Is that the pace you want for your game?

Maybe, maybe not. Perhaps you want these encounters to run a little faster. Or maybe you’re designing a high-level challenge, like a boss fight. Once you’ve nailed down these equations, it doesn’t really matter—you can make any adjustments necessary to either extend or shorten the time players spend engaged in a particular action.

Keep in mind, this formula isn’t restricted to combat scenarios. This math applies to any situation that requires the acquisition or depletion of a resource. You can call that resource HP, mana, psychic points, or something else entirely.

What matters most is what the players are actually doing. They’re rolling dice to try to achieve a goal and possibly avoid a negative outcome. How likely are they to succeed or fail? How long do they have to spend on that task before the outcome is revealed? As a game designer, these equations give you the power to answer those questions with confidence. Are you ready to build your own game? Math out your system mechanics? Leave a comment, I would love to hear from you!