Warhammer 40k Dice Probability Calculator

Master the odds on the tabletop with our advanced 40k dice calculator, designed to help you understand hit rolls, wound rolls, and save rolls for every critical decision.

40k Dice Probability Calculator

Number of Successes	Probability (%)	Cumulative Probability (%)

Detailed breakdown of success probabilities for each possible outcome.

What is a 40k Dice Calculator?

A 40k dice calculator is a specialized tool designed to help Warhammer 40,000 players and enthusiasts understand and predict the outcomes of their dice rolls. In the complex and often brutal tabletop wargame of Warhammer 40,000, dice rolls are fundamental to determining success or failure in everything from shooting attacks and melee combat to special abilities and saving throws. This calculator quantifies the statistical probabilities associated with these rolls, allowing players to make more informed strategic decisions, assess risk, and potentially improve their win rate.

Who should use it:

• New Players: To grasp the basic mechanics and understand why certain strategies might be more reliable.
• Experienced Players: To optimize army lists, fine-tune tactics, and understand the statistical edge provided by certain abilities or unit synergies.
• Hobbyists and Theorycrafters: To explore hypothetical scenarios, compare different unit loadouts, or simply satisfy curiosity about the game’s underlying probabilities.

Common Misconceptions:

• “It’s just luck”: While luck plays a role, understanding probabilities allows you to mitigate bad luck and capitalize on good fortune through strategic choices. A calculator helps reveal the true likelihood of events, moving beyond gut feelings.
• “Calculators are only for competitive players”: Anyone who plays Warhammer 40,000 can benefit. Knowing the odds helps in understanding why your unit performed well or poorly, leading to more enjoyable games.
• “All rolls are equal”: The calculator highlights how modifiers, special rules (like rerolls), and target numbers significantly alter probabilities, demonstrating that not all dice rolls are created equal.

40k Dice Calculator Formula and Mathematical Explanation

The foundation of the 40k dice calculator lies in probability mathematics, specifically binomial probability, adapted for the six-sided dice (d6) used in the game. The process involves several steps to determine the likelihood of achieving a desired outcome.

Core Probability Calculation

For a single die roll, the probability of success depends on the target number required and any applicable modifiers. A standard d6 has faces numbered 1 through 6.

Let T be the target number (e.g., 4 for a 4+).

Let M be the modifier (e.g., -1 for a penalty).

The Effective Target Number (ET) is calculated as: ET = T + M.

The number of successful outcomes on a single die is: Success Outcomes = 6 – ET + 1.

The probability of success on a single die (p) is: p = (6 – ET + 1) / 6. However, if ET is 7 or higher, the probability is 0. If ET is 1 or lower, the probability is 1 (or 100%).

The probability of failure on a single die (q) is: q = 1 – p.

Handling Rerolls

Rerolls complicate the calculation, as they effectively change the probability of success for dice that are rerolled.

1. Reroll Ones: If ‘Reroll Ones’ is selected, any die rolling a ‘1’ is rerolled. The probability of success on a rerolled ‘1’ is the probability of rolling a 2 through 6, which is 5/6. The new probability of success for dice subject to this rule is calculated by considering the outcomes: 1 (reroll) or 2-6 (success). If the original target number is 2 or more, rolling a 1 is a failure. With rerolls, a ‘1’ becomes a chance to succeed. The effective success probability becomes: p’ = p + (1/6) * (probability of success on a reroll). For ‘Reroll Ones’, this often means p’ = p + (1/6) * (5/6) if the original target was >1. A simpler way is to consider: original success (p), original failure and reroll (q * (5/6)). The new probability p_new = p + q*(5/6). If the target is 1, then 1 is already a success, and rerolling it doesn’t change anything regarding success, only that a 1 is no longer a failure if it’s rerolled. The most common rule is that a 1 is always a failure *unless* it is explicitly rerolled. If ‘Reroll Ones’ applies to Hit rolls and the BS is 4+, then a 1 is a failure. If rerolled, it can become a 2,3,4,5,6. A 4+ hit would succeed if it becomes 4,5,6. So the probability becomes: p_success_on_reroll = (6 – max(2, ET)) / 6. The overall probability: p_new = p + (1/6)*(p_success_on_reroll).
2. Reroll Failed Rolls: If ‘Reroll Failed Rolls’ is selected, any die that fails the target number is rerolled. The probability of success for a die that failed its first roll becomes the probability of success on a d6 (p). The new probability of success (p”) is calculated by considering: original success (p) OR (original failure AND successful reroll). p” = p + q * p.
3. Combined Rerolls: The most complex scenario involves both ‘Reroll Ones’ and ‘Reroll Failed Rolls’. The order matters. Typically, ‘Reroll Ones’ is applied first. Any ‘1’s are set aside and rerolled. Then, any dice that still failed (including those that were not ‘1’s initially, and those that became failures after rerolling ‘1’s) are subject to the ‘Reroll Failed Rolls’ rule.

Binomial Probability for Multiple Dice

Once the probability of success for a single die (p_effective) is determined after all rerolls, we can calculate the probability of achieving exactly ‘k’ successes out of ‘n’ dice using the binomial probability formula:

P(X=k) = C(n, k) * (p_effective)^k * (1 – p_effective)^(n – k)

Where:

• n = number of dice rolled
• k = number of successful outcomes
• p_effective = probability of success on a single die (after rerolls)
• C(n, k) = The binomial coefficient, “n choose k”, calculated as n! / (k! * (n-k)!)

Overall Probabilities

• Probability of At Least One Success: 1 – P(X=0) = 1 – (1 – p_effective)^n
• Probability of All Dice Failing: P(X=0) = (1 – p_effective)^n

Variables Table

Variable	Meaning	Unit	Typical Range
n (Number of Dice)	Total number of dice being rolled for a specific action.	Count	1 – 100+
T (Target Number)	The face value required on a die to count as a success.	d6 Face Value	1 – 6
M (Modifier)	Adjustment to the target number (positive or negative).	Integer	-6 to +6 (effectively)
ET (Effective Target)	The target number after applying the modifier.	d6 Face Value	1 – 7+
p (Base Success Prob)	Probability of success on a single die roll before rerolls.	Percentage / Decimal	0% – 100%
p_effective (Final Success Prob)	Probability of success on a single die roll after considering all reroll rules.	Percentage / Decimal	0% – 100%
q (Base Failure Prob)	Probability of failure on a single die roll before rerolls.	Percentage / Decimal	0% – 100%
k (Number of Successes)	The specific number of successful rolls desired.	Count	0 – n

Practical Examples (Real-World Use Cases)

Example 1: Standard Infantry Shooting Phase

Scenario: A squad of 5 Space Marines is shooting with Bolters. Their Ballistic Skill (BS) is 3+, meaning they need a 3 or higher to hit. They are not within range for any modifiers.

Inputs:

• Number of Dice: 5
• Target Number: 3
• Modifier: 0
• Reroll Ones?: No
• Reroll Failed?: No

Calculation:

• Base Success Probability (p): A target of 3+ means rolling 3, 4, 5, or 6. That’s 4 outcomes out of 6. So, p = 4/6 = 2/3 ≈ 66.67%.
• Probability per die = 66.67%
• Probability of failure per die = 1 – 66.67% = 33.33%
• Probability of all 5 dice failing = (33.33%)^5 ≈ 0.0043%
• Probability of at least one success = 1 – 0.0043% ≈ 99.9957%
• Main Result (Chance of hitting with at least one shot): ~99.99%

Interpretation: With a BS of 3+, it is almost guaranteed that at least one of the 5 Bolter shots will hit. The calculator can also show the probability of getting exactly 3 hits, 4 hits, etc., which is useful for calculating expected damage.

Example 2: A Powerful Monster’s Charge Move

Scenario: A Tyrannosaurus Rex charges. It needs to roll a 7+ on 2d6 to successfully make the charge. It has an ability allowing it to reroll any charge roll that failed.

Inputs:

• Number of Dice: 2
• Target Number: 7
• Modifier: 0
• Reroll Ones?: No
• Reroll Failed?: Yes (Charge Rolls)

Calculation:

First, let’s analyze the base probability of rolling a 7+ on 2d6. The minimum roll is 1+1=2, the maximum is 6+6=12. Possible sums range from 2 to 12. The sum of 7 can be achieved by (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) – 6 combinations. Total combinations for 2d6 is 6*6=36. So, base probability of rolling a 7 is 6/36 = 1/6 ≈ 16.67%.

Now, apply the reroll failed rolls rule. The probability of success on the first roll is p = 16.67%. The probability of failure is q = 1 – 16.67% = 83.33%.

The new probability of success (p_effective) after rerolling failures is: p_effective = p + q*p = 16.67% + (83.33% * 16.67%) ≈ 16.67% + 13.89% ≈ 30.56%.

Main Result (Chance of successfully charging): ~30.56%

Interpretation: While a 7+ on 2d6 seems daunting (only 16.67% chance initially), the ability to reroll failed charge rolls significantly boosts the success rate to over 30%. This information helps a player decide when to commit such a valuable unit to a charge.

How to Use This 40k Dice Calculator

Our 40k dice calculator is designed for ease of use, providing instant probability insights. Follow these steps to get the most out of it:

1. Identify the Roll Type: Determine what kind of roll you need to calculate. Is it a Hit Roll, a Wound Roll, a Saving Throw, or a Charge Roll? This will help you set the correct target number.
2. Determine the Number of Dice: Count the total number of dice you will be rolling for this specific action. For example, if your weapon fires 3 shots, you’ll roll 3 dice for hits.
3. Set the Target Number: Refer to your Warhammer 40,000 rules. Your unit’s Ballistic Skill (BS) or Weapon Skill (WS) determines the target number for hits. The target number for wounds is determined by comparing the Strength (S) of the attacker to the Toughness (T) of the target (e.g., S vs T, S = T -> 3+, S vs T -> 4+, etc.). Armour Saves (AS) determine the target number for successful saves. Charge rolls typically have a fixed target like 7+.
4. Apply Modifiers: Check for any rules that add or subtract from the target number. Examples include penalties for moving heavy weapons (-1 to hit), cover bonuses (+1 to saves), or certain psychic powers. Enter these as positive or negative numbers.
5. Input Reroll Abilities: Select the appropriate options if your unit or weapon has abilities to reroll dice. Specify if it’s rerolling ‘1’s, rerolling failed rolls, or both. Be precise about what type of roll (Hit, Wound, Save) the reroll applies to if specified by the rule.
6. View the Results: Once you’ve entered the inputs, the calculator will instantly update.
   • Main Result: This is typically the probability of achieving at least one success (e.g., at least one successful hit, wound, or save). This is often the most crucial number for overall success.
   • Intermediate Values: These provide a breakdown:
     • Chance of Success per Die: The final, effective probability of a single die rolling successfully after all modifiers and rerolls.
     • Chance of Failure per Die: The inverse of the success per die probability.
     • Chance of All Dice Failing: The probability that none of your dice achieve success.
     • Chance of At Least One Success: A crucial metric, often what the main result represents.
   • Formula Explanation: Understand the math behind the results.
   • Chart and Table: Visualize the probability distribution and see the likelihood of achieving any specific number of successes.

Decision-Making Guidance

• High Probability (e.g., >90% for at least one success): You can be highly confident in the outcome.
• Moderate Probability (e.g., 50-70%): The outcome is uncertain. Consider using stratagems or abilities to improve reliability if the result is critical.
• Low Probability (e.g., <30%): The outcome is unlikely. Relying on such a roll might be risky. Consider alternative strategies or units that offer better odds.

Use the “Copy Results” button to quickly share your calculated probabilities or save them for reference. The “Reset” button allows you to start fresh with default values.

Key Factors That Affect 40k Dice Calculator Results

Several elements in Warhammer 40,000 significantly influence the probabilities calculated by our dice calculator. Understanding these factors is key to leveraging the tool effectively:

1. Target Number (Core Difficulty): This is the most fundamental factor. A lower target number (e.g., 2+) offers a much higher probability of success per die than a higher number (e.g., 5+). Improving your Ballistic Skill, Weapon Skill, or finding ways to reduce the opponent’s save are primary ways to manipulate this.
2. Modifiers (Accuracy Adjustments): Modifiers can drastically swing probabilities. A -1 to hit or wound can turn a reliable attack into a less consistent one, while a +1 to save can make a flimsy unit surprisingly durable. Always account for these, as they directly alter the effective target number.
3. Reroll Abilities (Mitigating Failure): Rerolls are incredibly powerful. The ability to reroll ‘1’s or all failed rolls can significantly boost the chance of success, especially when rolling many dice or facing difficult target numbers. Units with access to these abilities are statistically more reliable.
4. Number of Dice Rolled (Volume of Fire/Attacks): While the probability of success *per die* might remain constant, rolling more dice increases the overall chance of achieving at least one success and also increases the expected number of successes. This is why units with many attacks or shots can be so devastating, even with less accurate weapons.
5. Special Rules (Ignoring/Modifying Outcomes): Some rules allow units to ignore cover, automatically wound certain targets, or grant automatic successes. These bypass the standard dice mechanics and often provide a near-guaranteed outcome, effectively having a 100% success rate for that specific interaction.
6. Dice Type: While standard d6 are used for most rolls, some special weapons or abilities might interact with other dice types (though less common in 40k core rules). The calculator is specifically for d6.
7. Player Decisions (When to Use Abilities): The calculator provides raw probability, but the player’s decision on when to use limited resources like Command Points for rerolls, or when to deploy specific units, is a meta-factor. The tool informs these decisions by quantifying the risk/reward.
8. Opponent’s Strategy: The opponent’s unit choices, positioning (e.g., granting cover), and available stratagems directly influence the target numbers and modifiers you’ll face, indirectly affecting the calculator’s relevance in a specific game situation.

Frequently Asked Questions (FAQ)

Q: What’s the difference between BS/WS and target numbers for wounding?

A: Ballistic Skill (BS) and Weapon Skill (WS) determine the target number needed to *hit* with ranged and melee attacks, respectively. Wound rolls have a target number determined by comparing the attacker’s Strength (S) characteristic against the defender’s Toughness (T) characteristic. For example, S=T requires a 4+ to wound.

Q: How do modifiers like Cover work?

A: Cover typically grants the target a bonus to their saving throw (e.g., +1 to Save). This means if the target has a 4+ save, cover makes it effectively a 3+ save (you need a 3 or higher on the die). The calculator handles this by adjusting the ‘Target Number’.

Q: Can I calculate the probability of a specific number of successes (e.g., exactly 3 hits)?

A: This calculator’s main result focuses on the probability of *at least one* success for simplicity and common use. However, the underlying binomial probability formula (explained in the article) allows for calculation of exact numbers of successes. The chart also visually represents this distribution.

Q: What if I have multiple reroll abilities?

A: The calculator allows selection for ‘Reroll Ones’ and ‘Reroll Failed Rolls’. If you have unique abilities (e.g., rerolling specific numbers other than 1), you may need to calculate manually or use a more advanced tool. The order of operations in the calculator assumes ‘Reroll Ones’ applies first, then ‘Reroll Failed Rolls’.

Q: Does the calculator account for critical hits/saves?

A: Standard critical hits/saves (like rolling a 6 being a critical success) are usually incorporated into the base target number if they are treated as a regular success (e.g., a 6 is still a success on a 6+ to hit). If a 6 grants a special *additional* effect beyond a normal success, that specific bonus is not calculated here, as it’s a separate mechanic.

Q: What does “Chance of All Dice Failing” tell me?

A: This is the probability that your entire group of dice results in no successes at all. A low value here means you’re very likely to get at least one success, while a high value indicates a significant risk of getting zero results.

Q: How accurate are these probabilities in a real game?

A: The probabilities are statistically accurate based on the rules entered. However, actual game outcomes can vary due to dice randomness. The calculator helps you understand the *expected* outcome and manage risk, not guarantee results.

Q: Can I use this for games other than Warhammer 40,000?

A: If another game uses six-sided dice (d6) and involves similar target number rolls with potential modifiers and rerolls, you can likely adapt this calculator. However, it’s specifically tailored for 40k mechanics.

Related Tools and Internal Resources

• Warhammer 40k Strategy Guides

  Explore detailed strategies for various factions and unit types to complement your dice-rolling success.

• Understanding Warhammer 40k Rules

  Deep dive into the core rules of Warhammer 40k, including hit, wound, and save roll mechanics.

• Army Building Optimizer

  A tool to help you build synergistic army lists, considering unit effectiveness and points costs.

• Warhammer 40k Mission Tactics

  Learn how to approach different game missions and objectives, maximizing your chances of victory.

• Wound Roll Probability Guide

  A focused guide on the intricacies of wound roll calculations and factors affecting them.

• Saving Throw Analysis Tool

  Analyze the effectiveness of different armour saves against various weapon types.