Mathhammer explained How to calculate expected damage in Warhammer 40,000 · 11th edition
Mathhammer (or "math hammer") means using probability to predict what your attacks will do before you roll. It answers the questions that win games: which unit should shoot that target? Is this upgrade worth the points? How many attacks do I need to kill it? This guide covers the core maths. The free mathhammer calculator does all of it for you, for any unit in the game.
🧮 Open the mathhammer calculator
The attack sequence in one line
Expected damage = attacks × P(hit) × P(wound) × P(save fails) × damage. Each step is a dice roll with a known chance of success, so the average result is just those chances multiplied together. An unmodified roll of 1 always fails and an unmodified 6 always succeeds, which is why no roll is ever better than 5 in 6 (83.3%).
1. Hit rolls
You hit on the attacker's BS (shooting) or WS (melee) or better. Re-rolling 1s turns a sixth of every result into another roll; re-rolling all failures gives a second chance at every miss.
| Hit on | Chance | Re-roll 1s | Re-roll all misses |
|---|---|---|---|
| 2+ | 83.3% | 97.2% | 97.2% |
| 3+ | 66.7% | 77.8% | 88.9% |
| 4+ | 50% | 58.3% | 75% |
| 5+ | 33.3% | 38.9% | 55.6% |
| 6+ | 16.7% | 19.4% | 30.6% |
2. Wound rolls
Compare the weapon's Strength with the target's Toughness:
| Strength vs Toughness | Wound on | Chance |
|---|---|---|
| Strength is at least double Toughness | 2+ | 83.3% |
| Strength greater than Toughness | 3+ | 66.7% |
| Strength equal to Toughness | 4+ | 50% |
| Strength less than Toughness | 5+ | 33.3% |
| Strength half Toughness or less | 6+ | 16.7% |
3. Saving throws
The target's save is worsened by the weapon's AP (a 3+ save against AP-1 becomes 4+). An invulnerable save ignores AP, and the defender uses whichever is better. The number that matters for damage is the chance the save fails:
| Save after AP | Chance it fails |
|---|---|
| 2+ | 16.7% |
| 3+ | 33.3% |
| 4+ | 50% |
| 5+ | 66.7% |
| 6+ | 83.3% |
| none | 100% |
4. Damage and models removed
Each failed save deals the weapon's Damage. Random damage averages out: D3 is 2, D6 is 3.5, D6+1 is 4.5. Damage that exceeds what a model has left is lost, so 3-damage shots into 2-wound models waste a point each time. That's why "wounds removed" and "models killed" are better yardsticks than raw damage, and why Feel No Pain (ignoring each point of damage on a roll) is so strong against big single hits.
Worked example
Five power-armoured troopers, each with 2 shots at BS 3+, Strength 4, AP-1, Damage 1, shoot a squad of ten small T3 creatures with a 5+ save and 1 wound each.
| Attacks | 5 models × 2 shots | 10 |
|---|---|---|
| Hits | 3+ to hit = 66.7% | 6.67 |
| Wounds | S4 vs T3 = 3+ = 66.7% | 4.44 |
| Failed saves | 5+ worsened by AP-1 = 6+, fails 83.3% | 3.70 |
| Models removed | 1 damage each, 1 wound each | 3.7 |
On average the volley removes about 3.7 models. But 22% of the time it removes 2 or fewer, and 12.1% of the time 6 or more; you can count on at least 3 three times out of four. The calculator shows that whole spread for any matchup.
Special rules that change the maths
- Sustained Hits X: each critical hit (usually a 6) scores X extra hits. Worth about X/6 extra hits per attack.
- Lethal Hits: critical hits wound automatically, skipping the wound roll. Best against high Toughness.
- Devastating Wounds: critical wounds can't be saved. Best against good armour and invulnerable saves.
- Anti-X Y+: against targets with keyword X, wound rolls of Y+ are critical wounds, which is huge when combined with Devastating Wounds.
- Twin-linked: re-roll wound rolls. Torrent: hits automatically. Blast: extra attacks against big units.
- +1/-1 modifiers: most are capped at +1 or -1 in total, so stacking two of the same modifier usually does nothing.
From one attack to a whole army
The same maths, repeated for every weapon and divided by points, tells you which units earn their cost. See the mathhammer rankings for every faction, or import your list and your opponent's into the army comparator to see what each of your units does to each of theirs.
Mathhammer FAQ
What is mathhammer?
Mathhammer (sometimes written math hammer) is using probability to work out what an attack will do on average in Warhammer 40,000: how many hits, wounds, failed saves, damage and models removed you should expect, before you roll a single die.
How do you calculate expected damage in 40k?
Multiply through the attack sequence: attacks × chance to hit × chance to wound × chance the save fails × damage per failed save. Special rules such as re-rolls, Sustained Hits, Lethal Hits and Devastating Wounds change individual steps.
Is average damage enough?
No. Averages hide the spread: a unit that averages 4 kills might wipe the target half the time and do almost nothing the rest. Tactical Reroll's calculator rolls each attack thousands of times to show the full distribution, including the result you can count on 75% of the time.
What does points efficiency mean?
Output divided by cost: for example wounds removed per 100 points against a given target, or how many hits it takes to destroy a unit per 100 points it costs. It lets you compare a 60-point unit with a 200-point unit fairly.
Is there a free mathhammer calculator for 11th edition?
Yes. Tactical Reroll is free, runs in your browser, uses 11th edition data, and can import army lists from New Recruit, BattleScribe and the Warhammer 40,000 app.
Rules summarised in our own words for the maths; the official Warhammer 40,000 core rules are the authority.