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Many a philosopher ask, “what is best in life?”

A wise man once answered, “a natural 20…”

Every player beams with excitement upon rolling a critical hit. And when they land at just the right moment—the climax of battle, when all hope is lost, when the BBEG looms, memories are made, campaigns are immortalized, and victory is won!

But for the game designer, a natural 20 means something different. For the designer, it’s a question, “how do crits effect game-play?”

Picture this: you’re in the middle of a heated battle, your party’s surrounded, and the villain’s monologuing about their evil plan. It’s your turn, and the stakes couldn’t be higher. You pick up your d20, give it a roll, and… bam—natural 20! The table erupts in cheers, and for a moment, you’re the hero who just flipped the script on this fight.

Every TTRPG designer wants this level of excitement at the table. They want players, game Masters, dungeon Masters, storytellers, adventurers, all of them, to cheer, to cry, and to lose themselves in the game. And a critical hit certainly and thoroughly captures the hearts and minds of everyone willing to roll the math rocks, even if it’s only for a moment.

But how do designers account for this when trying to “balance” a system?

In this article, we’re diving into the math (yes, math) behind critical hits and how they impact combat. Designers must at least consider how impactful a crit is on your players. The thrill, the excitement, the addictive dopamine rush!

This kind of mechanic gets gamers hooked.

But you need to understand how it affects the mechanics. How fast will enemies fall when the almighty 20 is rolled? How will that affect the pacing? Will combat become too easy? Where is the balance between undefeatable foes and uninteresting encounters?

This isn’t a 9^th level equation, but it is just as impactful as a well-placed fireball. Understand critical hit mechanics and make a balanced and exciting game!

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Step 1: Establishing Combat Encounter Baselines

Before calculating the impact of critical hits, it’s important to establish some standard assumptions about combat. In D&D 5e, most encounters are designed to last 3 rounds on average, and the baseline party consists of 4 player characters (PCs). We’re going to go with this as our template for multiple reasons.

3-Round Encounter Rule

This is a common design principle utilized in many games. On average, a typical encounter concludes within 3-rounds of combat. Meaning every player at the table has had 3 turns.

This pace generally maintains excitement. It’s not too long, so players still have plenty of opportunities to pursue other goals during a session. And it’s not too short, which may cheapen the encounter and leave it feeling unimpressive.

You could say that 3 is the Goldilocks number of rounds.

4-Player Party

This is the standard party size assumed when determining damage outputs and encounter difficulty in many systems, including D&D. Tabletop role-playing games are designed to be played at (you guessed it) the table.

As such, many games, even boardgames, plan their mechanics around the 3 to 5 player template. If you’re having people over to play a game, eat dinner, and relax, you’re probably looking at roughly 3 to 5 people hanging out.

Most games offer options for exceeding five or playing with as few as two. It’s been a tried and true industry-standard sense… Forever! So we’re going to cut the difference and set the baseline at 4.

Player Number Flexibility

By grounding our analysis in these assumptions, we ensure consistency in our calculations, but that doesn’t mean you can’t adjust and customize. These equations can be altered to suit whatever number of players or rounds you intend for your system.

Maybe you want more players with fewer rounds. Maybe you want a system which is primarily combat focused, no problem. When you look at the equation, change the number of players or rounds to fit your goals.

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Step 2: The Probability of a Critical Hit

The goal here is to figure out the chances that at least one character in the party will roll a natural 20 (a critical hit) during combat. We know that each player has a 5% chance to roll a natural 20 with a d20 die. So, how do we calculate the likelihood of seeing at least one critical hit in a given round of combat?

First, let’s break down the math into steps:

Calculate the chance of not rolling a critical hit for one player.

• The chance of not rolling a 20 is the other 19 numbers on the die, meaning 19 out of 20 chances (or 95%).

Extend that for all four players in the party.

• Since each player rolls their own dice, we need to calculate the odds of all 4 players failing to roll a natural 20.
• To do this, multiply the chance of not rolling a critical hit for each player (95% or 0.95) across all four players:

Chance of no crits in one round:

0.95×0.95×0.95×0.95=0.8145

This means there’s an 81.45% chance that none of the players roll a critical hit in a round.

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Step 3: Calculate the chance of 1 critical hit.

Now that we know how likely a critical hit is, let’s look at how it affects total damage. First, we will need a baseline. Just like the number of rounds in combat and number of players at the game table was flexible, the amount of damage done on each successful attack roll can be modified.

I’m giving you an equation. Alter the numbers within the equation to fit the system you’re trying to build. Whether you’re using damage dice, static damage number, or even the difference between an opposing attack and defense role, you can use the following equation.

The most common damage die in Dungeons & Dragons is a 1d8, so that is my baseline. You can easily switch this out with any other damage die. You can even find the average of all the different types of damage dice averages available in your system. Then simply place that overarching average into this equation.

The average score rolled on any dice is half the maximum number of possible +.50.

Which looks like…

Finding Base Damage

• Remember, damage is the damage dice plus any applicable modifier such as strength or dexterity. We will be using +3 as the added modifier.
• 4.5 ^{(1d8 average)} + 3 ^(modifier) = 7.5 ^{(average damage)}
• Over 3 rounds, with 4 players, this adds up to:

Base total damage:

4^(players) × 3^(rounds) × 7.5^{(average damage)} = 90 ^{(parties damage for around)}

Party’s Base total damage for three rounds: 4×3×7.5=90

So, without critical hits, the party is expected to deal 90 total damage in 3 rounds.

Critical Hit Damage

A critical hit doubles the damage dice but not the modifier, so if a player normally rolls 1d8 damage, a critical hit means they roll 2d8.

Therefore, The average damage for 2d8 + 3 crit is 12 instead of 7.5.

Factoring Critical Hits into Average Damage:

With an 18.55% chance of a critical hit each round, we can adjust the average damage per player’s attack. The equation balances the average damage of both a normal and a critical hit according to the probability of landing those types of hits.

Expected damage per attack:

(non-crit damage × non-crit attack roll) + (crit damage × crit attack roll)

For one player:

Expected damage = (7.5 × 0.815) + (12 × 0.185) = 8.35 This means, on average, each player deals about 8.35 damage per attack, factoring in critical hits.

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Step 4: Calculating Average Damage for the Entire Party Over Three Rounds

Now that we’ve calculated the expected average damage per attack for one player, let’s extend this to determine how much damage the entire party can deal in a typical combat encounter.

Calculating Damage per Round for 1 Player:

We previously calculated that each player, on average, deals 8.35 damage per attack, considering the chance of a critical hit.

Calculating for the Entire Party in One Round:

Since there are 4 players in the party, we multiply the average damage per player by 4:

Average damage per round for the party: 8.35^{(per player)} x 4^(players) = 33.4

This means that, on average, the party deals 33.4 damage per round, factoring in the probability of critical hits.

Average Damage Over a 3-Round Combat Encounter:

Since our goal for combat length is 3 rounds, we multiply the average party damage output by 3.

Total damage over 3 rounds:

33.4^{(damage per round)} × 3^(rounds) = 100.2

All of the Calculations and Equations

I walked through how and why all of these equations are put together. I’m going to list them out below with the baseline numbers we’ve already established. If you want to adjust any of those baselines, should be pretty easy for you to do so.

Customize the math to fit your game and goals.

1. Chance of No Critical Hits in One Round

Chance of no crits in one round = 0.95 × 0.95×0.95×0.95=0.8145

(81.45% chance that none of the players will roll a critical hit in a round)

2. Chance of at Least One Critical Hit in One Round

Chance of at least one crit  1− 0.8145 = 0.1855

(18.55% chance of rolling at least one critical hit per round)

3. Base Total Damage Without Critical Hits (for One Player)

Base total damage 4.5^{(1d8 average)} + 3^(modifier) = 7.5

4. Average Damage on a Critical Hit (for One Player)

Critical hit damage 2 × 4.5^{(2d8 average)} + 3^(modifier) = 12

5. Expected Damage Per Attack (Including Critical Hits)

Expected damage per attack = (7.5×0.815) ^{average non-crit} + (12×0.185)^{average crit} = 8.35 ^{average damage}

6. Average Damage Per Round for the Entire Party

Average damage per round for the party 8.35 ^{(per player)} × 4^(players) = 33.4

7. Total Expected Damage Over a 3-Round Encounter

Total damage over 3 rounds  = 33.4^{(damage per round)} × 3^(rounds) =100.2

Why a natural 20?

The critical hit is a fun mechanic that occurs sparingly but electrifies game day. It’s rarity makes its occasional appearance stunningly fun! It doesn’t have to overpower your system and it doesn’t even have to be restricted to a single number. It’s an excellent tool that any game designer should keep in their back pocket.

This math can be applied and adjusted in any number of ways. What matters most is that your goal is “fun”. These stats should help keep the pace and fun in line with your system.

Good luck designer!

And don’t forget to keep rolling them dice…