Introduction to Combat Maths, Part 1

The Basics

Instead of adding a lengthy parenthetical section in the fighter DPS article, it was decided a separate series of articles covering the combat calculations would be written. Any questions, comments, and critiques are welcomed. The essentials will be covered here.

Chance To Hit

Essential in applying damage to a target is an opportunity to hit it. It could be determined, say using fingers and toes, the number of sides of a d20 that represent success or failure of a roll for each target AC and given bonus to hit. Or, with shoes and socks in place, a generalizable equation could be used.

To quickly develop the equation, common arithmetic routinely used in combat should be considered. Given a target AC = 17 and a to hit bonus of +6, what roll do you need to hit? An 11 and above on a d20 will hit. Since there are only 20 possible outcomes in any single roll of a d20 and 1-10 will miss, the chance to miss 50%. The chance to hit is 100% less the chance to miss.

This method can be generalized to an equation:

And the average chance to hit is generalized to the complement of this as:

The decimal value can be multiplied by 100 to represent the % value if desired. Note also that there are boundaries not represented in equation but present in game rules, namely there is without advantage a 5% miss chance and at least a 5% critical hit chance represented by values on the d20. Last, the reason to use chance to miss will become apparent in Part 2.


Similarly, the basic form of the damage calculation is not unlike that commonly used at the table. First, after an attack is successful, one determines the damage given by weapon die. Then the weapon’s ability modifier is added. Next, any applicable bonuses are added. Damage on a hit is given, with ability modifier represented by M and other bonuses represented by B, by the equation:

Over time, the average weapon rolls of a weapon will approach the sum of its weapon die’s average values. For example, a great sword has 2d6. Each d6 has an average value of halfway between its maximum of 6 and its minimum of 1, or 3.5. These dice are summed to find the average weapon damage of a great sword at 7 damage. Similar reasoning can be used to discover the average weapon damage of any weapon.

Critical Hit Chance and Damage

Unlike the hit chance, the critical hit chance does not depend on AC of opponent. Thus for any given chance to critical and weapon die, the contribution to average damage over time is constant. In other words, for a critical hit chance of 5%, 1 in 20 of all rolls on average over time will be a critical hit whether those swings are at a coat rack or at Her Dark Majesty Tiamat, Queen of Dragons. The average extra damage beyond a guaranteed hit is given by equation:

Average Damage Per Turn of Combat

Bringing it all together, average combat damage for n attacks is represented as:

Keeping in mind there are bounds on chance to hit without advantage due to a roll of 1 on a d20 always being a miss and a roll in the range of a critical always being a hit, given by:

Part 2 will cover advantage and its relation to critical hits or misses.

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