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 40k damage calculator does all of it for you, for any unit in the game.

Simulate itIntercessor Squad vs Terminator Squad›

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.

Try it: 3+ to hit ›Re-rolling 1s ›Re-rolling every miss ›

2. Wound rolls

Compare the weapon's Strength with the target's Toughness:

S5 into T3 ›S5 into T5 ›S5 into T6 ›

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:

AP-1 into a 3+ save ›Into a 2+ save with a 4++ ›

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.

How modifiers stack in 11th edition

Most arguments about the maths are really about modifiers. These are the points that change the numbers, in the order the dice see them:

• Re-rolls come first. A re-roll happens before any modifier is applied, and a die is never re-rolled twice. A re-rolled 6 still counts as a critical.
• Unmodified 1s fail and unmodified 6s succeed on hit and wound rolls, whatever the modifiers say. That is why nothing ever hits better than 2+ or worse than 6+.
• Changes to a skill change the skill itself. When BS or WS gets better or worse (cover worsens the BS of ranged attacks by 1 in 11th), the characteristic moves, not the roll, and it can't go past 2+ or 6+. The calculator keeps these apart from the +1/−1 roll modifiers.
• Roll modifiers to hit and to wound are held to +1 or −1 in total. Two +1s to hit give +1, and a +1 and a −1 cancel out; that's why a second buff of the same kind is often wasted.
• AP and saves. AP makes the armour save worse; worsening a weapon's AP stops at 0. A save can't be better than 2+. An invulnerable save ignores AP entirely, and the defender uses whichever save is better after AP.
• Damage changes come after the save. Rules that reduce or halve damage apply to an attack that got through, and damage never drops below 1 unless a rule says otherwise. Feel No Pain rolls come after that, one per wound lost.

Characters leading squads

In 11th edition a character can join a squad and they fight as one unit. Attacks against it use the squad's Toughness while any of the squad is left. All the save dice are then rolled together and worked through from the lowest up, each against whoever stands at the front: the squad first, the character last. So the character usually only meets the highest dice, which are mostly saves, unless the weapon has Precision, which puts the character at the front. The calculator plays this out roll by roll: add a leader to either side with ＋ Leader, or import a list that says who's attached.

Try it: a squad led by a character ›

Worked example

Five power-armoured troopers, each with 2 shots at BS 3+, Strength 5, AP-1, Damage 1, plus 1 more each within half range (Rapid Fire 1), shoot a squad of ten small T3 creatures with a 5+ save and 1 wound each.

On average the volley removes about 5.5 models. But 13.4% of the time it removes 3 or fewer, and 6% of the time 9 or more; you can count on at least 4 three times out of four. The calculator shows that whole spread for any matchup.

Try the worked example ›

Worked examples from our engine

The same sums, done by the engine behind the calculator and our unit pages, on real units with their best squads: standing still, no cover, beyond half range unless the row says otherwise. These numbers are worked out from the current data each time the site is built.

Attack	Attacks	Hits	Wounds	Failed saves	Damage	Models removed	Try it ›
5 Terminator Squad fight 5 Marines	15	10.00	8.33	5.56	9.04	4.52
10 Cadian Shock Troops shoot 5 Marines	13.50	8.50	3.00	1.22	1.38	0.53

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. Try it ›
• Lethal Hits: critical hits wound automatically, skipping the wound roll. Best against high Toughness. Try it ›
• Devastating Wounds: critical wounds can't be saved. Best against good armour and invulnerable saves. Try it ›
• 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.

Common mistakes

• Averaging dice wrong. A D6 averages 3.5, not 3, and a D3 averages 2. Small slips add up over a whole army.
• Counting damage a model can't take. Damage past a model's last wound is lost, so a 3-damage weapon into 2-wound models only ever does 2 per hit that gets through.
• Stacking modifiers past the cap. A second +1 to hit usually does nothing; spend that buff where it changes a number.
• Forgetting the invulnerable save. High AP stops mattering once it reaches the target's invulnerable save.
• Treating the average as a promise. An average of 4 kills can mean a wipe half the time and almost nothing the rest. Look at the spread and the result you can count on.
• Comparing raw damage instead of damage per point. A 250-point unit should out-damage a 90-point one; the question is whether it does so per point, into the targets you'll meet.
• Applying cover in melee. The benefit of cover only affects ranged attacks.

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 tier lists for every army, or load your list and your opponent's into the calculator (Pick a list on each side) to test your units against theirs.

SimulateCompare two army lists

Words we use

The site's own terms, each defined in a line on the Method page: Share slain · Wounds removed per 100 points · Points removed · Benchmark targets · Best loadout · All-round loadout · Letter · Dashed letter · SWORD · SHIELD · Role · 9 in 10, average, 1 in 10.

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.

Does cover still add 1 to the save in 11th edition?

No. In 11th edition a unit with the benefit of cover makes ranged attacks against it worse at hitting: the attack's Ballistic Skill is worsened by 1. It doesn't affect melee.

Do +1 to hit modifiers stack in 40k?

Only up to a point: the total modifier to a hit roll, or to a wound roll, is held to +1 or −1. Improving the Ballistic or Weapon Skill itself is a separate change to the characteristic.

What's the difference between Lethal Hits and Devastating Wounds?

Lethal Hits skips the wound roll on a critical hit, which helps most against tough targets. Devastating Wounds skips the save on a critical wound, which helps most against good armour and invulnerable saves.

How many attacks does it take to kill a unit?

Divide the wounds you need to remove by the damage one attack does on average: chance to hit × chance to wound × chance the save fails × damage per attack, counting damage lost past each model's last wound. The calculator does this for any pair of units.

Rules summarised in our own words for the maths; the official Warhammer 40,000 core rules are the authority.

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