Using probability theory to help optimizing damage

[Oerlaf]

I have made a research on single-target damage dealing spells of three types: hit-or-miss, save-or-damage, and save-on-half, and I have obtained interesting results.

Short version: if a spell deals NdX damage, the base expected damage is N(X+1)/2 (hereafter BED)

I. Hit-or-miss spells.
With a fixed to hit modifier each point of AC additively reduces the expected incoming damage by 5% (up to a minimum of 10% from BED).

With a fixed AC each point of tohit additively increases the expected incoming damage by 5% (up to a maximum of 100% from BED)

2. Save-ot-damage spells
With a fixed DC each point of the relevant ability modifier (Abi) reduces the expected incoming damage by 5% (up to a minimum of 0% from BED)

With a fixed relevant ability modifier each point of DC increases the expected incoming damage by 5% (up to a maximum of 100% from BED)

2. Save-on-half spells
With a fixed DC each point of the relevant ability modifier reduces the expected incoming damage by 5% (up to a minimum of 50% from BED)

With a fixed relevant ability modifier each point of DC increases the expected incoming damage by 5% (up to a maximum of 100% from BED)


1. Hit-or-miss.
Let z = AC - tohit. The expected incoming damage can be calculated as follows:

If z<2 - N(X+1)/2;
If 2 <= z <= 20 - N(X+1)/2*(22-z)/20;
If z > 20 - N(X+1)/20

2. Save-or-damage
Let z = DC - Abi.
The expected incoming damage can be calculated as follows:

If z<1 - 0;
If 1 <= z <= 21 - N(X+1)/2*(z-1)/20;
If z > 21 - N(X+1)/2

2. Save-on-half
Let z = DC - Abi.
The expected incoming damage can be calculated as follows:

If z<1 - N(X+1)/4;
If 1 <= z <= 21 - N(X+1)/4*(z+19)/20;
If z > 21 - N(X+1)/2


Why have I done it and what is it useful for? When we play D&D we know two of the four variables presented here: our tohit modifier and our spell save DC. The other two variables can often be predicted. Thus, we can each round select the spell that is going to deal the highest amount of expected damage.

[Tiadoppler]

Make sure to factor in a target's damage resistance (if applicable) and any secondary effects a spell may have. Straight-up single target damage is rarely the point of casting a spell.

My preference for spellcasters is to focus on:

AoE damage or effects. They have a multiplicative effect on larger encounters compared to single target damage.
Save or Suck against single, tough opponents, targeting the lowest save possible (keep in mind, enemies will all have 6 different saves to look at, not just one).
Buffs and heals for allies.

[JNAProductions]

What about efficiency? You can, pretty easily, determine that, say, Cone Of Cold does the most damage to the horde of goblins gnawing at you. But is it an efficient use of slots?

[Merudo]

Not sure why you used these non-straightforward formulas.

Reducing the hit rate by 5% will of course reduce the expected damage by 5%, since Expected damage = Hit rate * Damage when hit

Also, the calculations for AC are incorrect as they don't take into account criticals.

[Oerlaf]

Not sure why you used these non-straightforward formulas.

Reducing the hit rate by 5% will of course reduce the expected damage by 5%, since Expected damage = Hit rate * Damage when hit

Also, the calculations for AC are incorrect as they don't take into account criticals.

The formula under spoiler tag takes thise into account