Infinity Dice Calculator

Analyze exploding dice probabilities and expected values for any tabletop system.

What is an Infinity Dice Calculator?

An infinity dice calculator is a specialized statistical tool used by gamers, game designers, and mathematicians to model “exploding dice.” In many tabletop role-playing games (RPGs) like Shadowrun, Savage Worlds, or World of Darkness, rolling the maximum value on a die allows you to roll again and add the new result to the total. This process can theoretically repeat forever, hence the term “infinity dice.”

Using an infinity dice calculator helps players understand the true power of their characters and assists GMs in balancing encounters. Unlike standard dice where the average is simply the midpoint, the infinity dice calculator accounts for the recursive nature of explosions, which skews the average significantly higher.

Infinity Dice Calculator Formula and Mathematical Explanation

The math behind an infinity dice calculator relies on infinite geometric series. When a die “explodes,” the expected value (EV) increases because there is a non-zero chance of adding more value infinitely.

The Core Formula:

EV_total = (EV_standard) / (1 – P_explode)

Variable	Meaning	Unit	Typical Range
S	Sides on the die	Integer	2 – 100
N	Number of dice rolled	Integer	1 – 50
P_explode	Probability of rolling the explode value	Decimal	0.01 – 0.50
EV_standard	Expected value of a non-exploding die	Value	1.5 – 50.5

Practical Examples (Real-World Use Cases)

Example 1: Savage Worlds Wild Die (d6)

In Savage Worlds, a Wild Die is a d6 that explodes on a 6. Using the infinity dice calculator:

• Input: 1 Die, 6 Sides, Explode on 6.
• Base EV: 3.5.
• Calculation: 3.5 / (1 – 1/6) = 3.5 / 0.833 = 4.2.
• Interpretation: Over thousands of rolls, your average result will be 4.2 rather than the standard 3.5.

Example 2: Shadowrun Pool (10d6)

If you are rolling 10 dice where every 6 explodes:

• Input: 10 Dice, 6 Sides, Explode on 6.
• Infinity Dice Calculator Output: 42.0 Total EV.
• Interpretation: The “infinity” mechanic adds 7 full points of expected damage/successes compared to standard dice.

How to Use This Infinity Dice Calculator

To get the most accurate results from this infinity dice calculator, follow these steps:

1. Enter Number of Dice: Input the total count of dice in your pool.
2. Define Die Sides: Enter the faces of the dice (e.g., 20 for a d20).
3. Set Explosion Threshold: Most games explode on the highest number, but some systems allow explosions on other values. Enter that value here.
4. Analyze the Results: Review the primary Expected Value and the standard deviation to understand the “swinginess” of your roll.
5. Check the Chart: Use the visual representation to see how much the infinity dice calculator adjustment shifts the mean compared to standard mechanics.

Key Factors That Affect Infinity Dice Calculator Results

• Die Size: Smaller dice explode more frequently (a d4 explodes 25% of the time, while a d20 only 5%). This makes the infinity dice calculator impact much higher on smaller dice.
• Number of Dice: As the pool grows, the variance increases, leading to higher potential peaks.
• Explosion Threshold: Lowering the threshold (e.g., exploding on 5 or 6) dramatically increases the Expected Value.
• Reroll Limits: Some systems limit explosions to 3 times. Our infinity dice calculator assumes true infinity, which is the mathematical ceiling.
• Mathematical Convergence: The series only converges if the probability of exploding is less than 1.
• Game Balance: Developers use an infinity dice calculator to ensure that high-variance weapons don’t break the game’s math.

Frequently Asked Questions (FAQ)

Does a d4 with exploding dice have a higher average than a d6 without?

Yes. A standard d6 has an EV of 3.5. An exploding d4 has an EV of 3.33. Wait, they are close! However, a d6 that explodes (4.2) is significantly better. Use the infinity dice calculator to compare specific edge cases.

Can an infinity dice calculator predict a critical hit?

It predicts the average outcome over time, not a specific roll. It calculates the statistical “weight” of the explosion mechanic.

What is the “swinginess” in an infinity dice calculator?

Swinginess refers to Variance. Exploding dice have much higher variance than standard dice, meaning you are more likely to get extremely high or low results.

Is “Exploding” the same as “Aces”?

In Savage Worlds, yes. “Acing” is the specific term for what the infinity dice calculator models.

Why is my EV so high for a d2?

A d2 (coin flip) that explodes on a 2 has an explosion probability of 50%. This doubles the standard EV, as calculated by the infinity dice calculator.

How does this apply to the World of Darkness 10-again rule?

The 10-again rule is a classic application for the infinity dice calculator. It calculates how many additional successes a 10 will generate on average.

Does the infinity dice calculator handle “penetrating” dice?

Penetrating dice (where you subtract 1 from rerolls) require a slightly different formula, but this tool provides the upper bound of that value.

What if multiple numbers explode?

If both 5 and 6 explode on a d6, the P_explode becomes 2/6 (33.3%). You can adjust the infinity dice calculator by adjusting the threshold logic.

Related Tools and Internal Resources

• Dice Roller Pro: A comprehensive tool for simulating thousands of random rolls.
• RPG Combat Simulator: Calculate damage per round using infinity dice calculator logic.
• Probability Distribution Tool: Visualize Bell Curves and Poisson distributions.
• Gaming Stat Helper: A library of formulas for tabletop game balance.
• Random Number Generator: Generate high-entropy random sequences.
• Tabletop Strategy Guide: Learn the best ways to utilize exploding dice mechanics.