Warhammer Math Calculator

Optimize Your Warhammer 40,000 Damage & Probability

Warhammer Math Calculator

Use this Warhammer Math Calculator to quickly determine the probabilities of hitting, wounding, and saving, as well as the expected total damage your units will inflict in Warhammer 40,000.

Detailed Probability Breakdown

Phase	Target Roll	Probability (Success)	Probability (Failure)
Hit Roll	N/A	0.00%	0.00%
Wound Roll	N/A	0.00%	0.00%
Save Roll	N/A	0.00%	0.00%
Feel No Pain	N/A	0.00%	0.00%

What is a Warhammer Math Calculator?

A Warhammer Math Calculator is an essential tool for players of tabletop wargames like Warhammer 40,000. It helps you quantify the statistical outcomes of combat interactions between units. Instead of relying on gut feelings or rough estimates, this calculator provides precise probabilities for hitting, wounding, saving, and ultimately, the expected damage output of an attack sequence. It breaks down the complex series of dice rolls into understandable percentages and average damage figures.

Who Should Use a Warhammer Math Calculator?

• Competitive Players: To optimize army lists, identify efficient unit matchups, and make informed tactical decisions during a game.
• List Builders: To compare the damage output of different weapon options or unit compositions before a game even starts.
• New Players: To understand the core mechanics of the game and how various stats (Strength, Toughness, AP, Saves) interact.
• Hobbyists: To satisfy curiosity about specific combat scenarios or to theory-craft new strategies.

Common Misconceptions about Warhammer Math

Many players underestimate or overestimate the impact of certain modifiers. For example, a +1 to hit or wound might seem minor, but its cumulative effect across multiple attacks can be significant. Similarly, the interaction between Armor Penetration (AP) and Invulnerable Saves is often misunderstood, leading to incorrect assumptions about a unit’s durability. A Warhammer Math Calculator clarifies these interactions, showing the true statistical advantage or disadvantage.

Warhammer Math Calculator Formula and Mathematical Explanation

The core of the Warhammer Math Calculator lies in calculating the probability of success for each stage of an attack sequence. Each stage involves rolling a D6 (six-sided die) and achieving a result equal to or greater than a target number. The probability of success for a target number ‘X+’ on a D6 is (7 – X) / 6. The probability of failure is (X – 1) / 6.

Step-by-Step Derivation:

1. Probability to Hit (P_Hit):
   • Determine the modified target to hit: BS/WS + Hit Modifier.
   • Clamp the target between 2 and 6 (a 1+ always hits, a 7+ never hits).
   • P_Hit = (7 - Modified Target to Hit) / 6.
2. Probability to Wound (P_Wound):
   • Determine the base target to wound based on Strength (S) vs. Toughness (T):
     • S ≥ 2T: 2+
     • S > T: 3+
     • S = T: 4+
     • S < T: 5+
     • S ≤ 0.5T: 6+
   • Apply the Wound Modifier: Base Target to Wound + Wound Modifier.
   • Clamp the target between 2 and 6.
   • P_Wound = (7 - Modified Target to Wound) / 6.
3. Probability to Fail Save (P_FailSave):
   • Calculate the effective Armor Save: Armor Save - |AP|.
   • Calculate the effective Invulnerable Save: Invulnerable Save (if applicable).
   • Determine the probability of failing the Armor Save: P_FailArmor = (Effective Armor Save - 1) / 6.
   • Determine the probability of failing the Invulnerable Save: P_FailInvuln = (Invulnerable Save - 1) / 6.
   • The defender uses the better save, so P_FailSave = MIN(P_FailArmor, P_FailInvuln) (if Invulnerable Save is present, otherwise just P_FailArmor). Clamp between 0 and 1.
4. Probability to Fail Feel No Pain (P_FailFNP):
   • If FNP is 0, P_FailFNP = 1 (no FNP, so always fail to ignore damage).
   • Otherwise, P_FailFNP = (FNP - 1) / 6. Clamp between 0 and 1.
5. Expected Total Damage:
   • Expected Total Damage = Attacks × P_Hit × P_Wound × P_FailSave × P_FailFNP × Damage per Wound.

Variable Explanations:

Key Variables for Warhammer Math Calculations

Variable	Meaning	Unit	Typical Range
Attacks	Number of attacks made by the unit/weapon.	Count	1 – 20+
BS/WS	Ballistic Skill (ranged) or Weapon Skill (melee) target.	D6+	2+ to 6+
Hit Modifier	Modifier applied to the hit roll.	+/-	-1 to +1 (often capped)
Strength (S)	Attacker’s Strength characteristic.	Value	1 to 14+
Toughness (T)	Defender’s Toughness characteristic.	Value	1 to 14+
Wound Modifier	Modifier applied to the wound roll.	+/-	-1 to +1 (often capped)
AP	Armor Penetration value of the weapon.	-Value	0 to -5
Armor Save (Sv)	Defender’s base armor save characteristic.	D6+	2+ to 6+
Invulnerable Save (Invuln Sv)	Defender’s invulnerable save characteristic.	D6+	0 (none) to 6+
Feel No Pain (FNP)	Defender’s ability to ignore wounds.	D6+	0 (none) to 6+
Damage (D)	Damage inflicted per successful wound.	Value	1 to 6+

Practical Examples (Real-World Use Cases)

Understanding the Warhammer Math Calculator in action helps in strategic planning.

Example 1: Comparing Anti-Infantry Weapons

Imagine you have two units, both firing 10 shots at a standard Space Marine unit (T4, Sv3+). Both weapons have S4, but Weapon A has AP-1 D1, and Weapon B has AP-0 D2.

• Weapon A (AP-1 D1):
  • Attacks: 10, BS: 3+, Hit Mod: 0
  • Strength: 4, Toughness: 4 (Wound on 4+), Wound Mod: 0
  • AP: -1, Save: 3+, Invuln: 0, FNP: 0, Damage: 1
  • Calculator Output: P(Hit) = 66.67%, P(Wound) = 50.00%, P(Fail Save) = 50.00% (3+ becomes 4+ due to AP-1), P(Fail FNP) = 100%. Expected Total Damage: 10 * 0.6667 * 0.5 * 0.5 * 1 * 1 = 1.67
• Weapon B (AP-0 D2):
  • Attacks: 10, BS: 3+, Hit Mod: 0
  • Strength: 4, Toughness: 4 (Wound on 4+), Wound Mod: 0
  • AP: 0, Save: 3+, Invuln: 0, FNP: 0, Damage: 2
  • Calculator Output: P(Hit) = 66.67%, P(Wound) = 50.00%, P(Fail Save) = 33.33% (3+ save), P(Fail FNP) = 100%. Expected Total Damage: 10 * 0.6667 * 0.5 * 0.3333 * 1 * 2 = 2.22

Interpretation: Even though Weapon A has AP-1, Weapon B with D2 is more effective against a 3+ save target without an invulnerable save, due to its higher damage per wound. This highlights how a Warhammer Math Calculator can reveal non-obvious optimal choices.

Example 2: Attacking a Resilient Monster

Consider a unit of 5 attacks (BS3+, S8, AP-2, D3) targeting a monster (T10, Sv2+, Invuln4+, FNP5+).

• Attacks: 5, BS: 3+, Hit Mod: 0
• Strength: 8, Toughness: 10 (Wound on 5+), Wound Mod: 0
• AP: -2, Save: 2+, Invuln: 4+, FNP: 5+, Damage: 3
• Calculator Output:
  • P(Hit) = 66.67% (3+)
  • P(Wound) = 33.33% (5+)
  • Effective Armor Save: 2+ becomes 4+ due to AP-2. P(Fail Armor) = 33.33%.
  • Invulnerable Save: 4+. P(Fail Invuln) = 33.33%.
  • P(Fail Save) = MIN(33.33%, 33.33%) = 33.33%
  • P(Fail FNP) = 66.67% (5+ FNP)
  • Expected Total Damage: 5 * 0.6667 * 0.3333 * 0.3333 * 0.6667 * 3 = 0.74

Interpretation: Despite high Strength and AP, the monster’s high Toughness, good saves, and FNP significantly reduce the expected damage. This shows that even powerful attacks can struggle against highly resilient targets, emphasizing the importance of focused fire or specific anti-monster weapons. This Warhammer Math Calculator helps you understand why certain units are so tough to crack.

How to Use This Warhammer Math Calculator

Our Warhammer Math Calculator is designed for ease of use, providing instant feedback on your combat scenarios.

1. Input Your Attacker’s Stats:
   • Number of Attacks: How many dice are rolled in the hit phase.
   • Ballistic Skill (BS) / Weapon Skill (WS): The target number for your hit roll (e.g., 3 for 3+).
   • Hit Roll Modifier: Any +/- modifiers to the hit roll (e.g., -1 for Heavy, +1 for Sustained Hits).
   • Attacker’s Strength (S): Your unit’s Strength characteristic.
2. Input Your Defender’s Stats:
   • Defender’s Toughness (T): The enemy unit’s Toughness characteristic.
   • Wound Roll Modifier: Any +/- modifiers to the wound roll.
   • Armor Penetration (AP): The weapon’s AP value (enter as a negative number, e.g., -1).
   • Defender’s Armor Save (Sv): The enemy’s base armor save (e.g., 3 for 3+).
   • Defender’s Invulnerable Save (Invuln Sv): The enemy’s invulnerable save (enter 0 if none, e.g., 4 for 4+).
   • Defender’s Feel No Pain (FNP): The enemy’s FNP save (enter 0 if none, e.g., 5 for 5+).
   • Damage per Wound (D): The damage value of your weapon.
3. Read the Results:
   • Expected Total Damage: The primary highlighted result shows the average damage you can expect to inflict.
   • Intermediate Probabilities: See the percentage chance for each phase: Probability to Hit, Probability to Wound, Probability to Fail Save, and Probability to Fail FNP.
   • Detailed Probability Table: Provides the exact target roll and success/failure probabilities for each phase.
   • Expected Damage Chart: Visualizes how your expected damage changes against different Toughness values, with and without FNP.
4. Decision-Making Guidance: Use these numbers to compare weapon effectiveness, evaluate unit matchups, and refine your army list. For instance, if a weapon has low expected damage against a specific target, consider alternative units or strategies. This Warhammer Math Calculator empowers you to make data-driven decisions.

Key Factors That Affect Warhammer Math Calculator Results

Several interconnected factors significantly influence the outcomes calculated by a Warhammer Math Calculator. Understanding these helps in optimizing your gameplay.

1. Number of Attacks: This is the most straightforward factor. More attacks mean more dice rolled, directly increasing the potential for hits and wounds. Even small improvements in hit/wound probability are amplified by a high volume of attacks.
2. Ballistic Skill (BS) / Weapon Skill (WS) & Hit Modifiers: The base skill determines how often you hit. Modifiers (like +1 to hit from an aura or -1 from a heavy weapon) have a profound impact. Changing a 4+ to a 3+ hit roll increases success probability from 50% to 66.67%, a 33% relative increase in hits.
3. Strength (S) vs. Toughness (T) & Wound Modifiers: This is a critical interaction. Wounding on a 2+ (S >= 2T) is vastly superior to wounding on a 6+ (S <= 0.5T). Even a single point of difference in S or T can shift the wound roll by an entire step (e.g., 3+ to 4+), drastically altering expected damage. Wound modifiers further fine-tune this.
4. Armor Penetration (AP) & Saves (Sv, Invuln Sv): AP directly reduces the effectiveness of the defender’s armor save. A high AP weapon can force even heavily armored units onto their invulnerable save or no save at all. The presence of an Invulnerable Save is crucial, as it ignores AP, making units with good invulnerable saves highly resilient against high-AP attacks. The Warhammer Math Calculator helps you see this interaction clearly.
5. Damage per Wound (D): While probabilities determine how many wounds get through, the damage characteristic determines the impact of each wound. A D3 or D6 weapon can quickly eliminate multi-wound models, even if fewer wounds get through. Balancing high probability with high damage is key.
6. Feel No Pain (FNP): FNP is a last line of defense, allowing a unit to ignore allocated wounds. It effectively acts as a damage reduction multiplier. A 5+ FNP means 1/3 of wounds are ignored, significantly increasing a unit’s effective durability. This is a powerful defensive ability that the Warhammer Math Calculator accounts for.

Frequently Asked Questions (FAQ) about the Warhammer Math Calculator

Q: What edition of Warhammer 40,000 does this Warhammer Math Calculator apply to?

A: This calculator is primarily designed for Warhammer 40,000 10th Edition rules, which feature standardized hit, wound, save, and Feel No Pain mechanics. While the core probability principles apply to other editions or wargames, specific rules (like “to hit” caps or wound roll modifiers) are based on 10th Edition conventions.

Q: Why is my expected damage so low even with many attacks?

A: Low expected damage often results from a combination of factors. Check if your unit is hitting on a poor BS/WS (e.g., 4+ or 5+), wounding on a high target (e.g., 5+ or 6+ against high Toughness), or if the defender has a strong save (especially an Invulnerable Save) and/or a Feel No Pain ability. Each failed roll significantly reduces the final damage. The Warhammer Math Calculator helps pinpoint where the bottleneck is.

Q: How do I account for abilities like “Sustained Hits” or “Lethal Hits”?

A: This basic Warhammer Math Calculator does not directly model complex abilities like Sustained Hits (additional hits on critical hits) or Lethal Hits (auto-wounds on critical hits). For these, you would need a more advanced calculator or perform separate calculations for the critical hit portion and add them to the base results. However, you can often approximate by adjusting the “Number of Attacks” or “Wound Modifier” for a rough estimate.

Q: What if a unit has multiple weapon profiles?

A: If a unit has multiple weapons with different profiles, you should run the Warhammer Math Calculator for each weapon profile separately and then sum the expected damage results to get the total expected damage for the unit.

Q: Can this calculator help me build my army list?

A: Absolutely! By comparing the expected damage output of different units and weapon loadouts against common enemy profiles (e.g., infantry, elite infantry, vehicles, monsters), you can make informed decisions about which units are most efficient for your desired role. It’s a powerful tool for optimizing your army list and understanding unit efficiency.

Q: How does the calculator handle modifiers that cap at +/-1?

A: The calculator applies modifiers directly to the target roll. In Warhammer 40,000 10th Edition, hit and wound rolls are typically capped at +/-1. While the calculator allows you to input larger modifiers, remember that in-game rules will cap their effect. For accurate results, ensure your input modifiers reflect the actual capped value.

Q: What is the difference between Armor Save and Invulnerable Save in the calculator?

A: The calculator automatically determines the best save to use. Armor Save is modified by Armor Penetration (AP), while Invulnerable Save ignores AP. If a unit has both, the player chooses which save to attempt. The calculator assumes the player will always choose the save that has a higher chance of success (i.e., lower probability of failure).

Q: Is this Warhammer Math Calculator useful for other wargames?

A: The underlying principles of dice probability (D6 rolls, target numbers) are common across many wargames. While the specific terms (BS, WS, AP, FNP) are Warhammer 40,000 specific, you can adapt the inputs to calculate probabilities for other systems that use similar D6 mechanics for attack, defense, and damage rolls.

Related Tools and Internal Resources

Enhance your Warhammer 40,000 experience with these related tools and guides:

• Warhammer Unit Efficiency Guide: Learn how to maximize the effectiveness of your units on the battlefield.
• 40k Army List Builder: Plan and optimize your army compositions with our comprehensive list builder.
• Dice Roll Probability Tool: A general tool for calculating probabilities of various dice rolls, useful beyond Warhammer.
• Wargaming Tactics Explained: Dive deeper into strategic concepts and battlefield maneuvers.
• Advanced Warhammer Rules: Explore complex rule interactions and edge cases to gain a competitive edge.
• Beginner 40k Guide: A perfect starting point for new players looking to understand the basics of Warhammer 40,000.