How are these probabilities computed?

By exact enumeration, not random simulation. The tool builds the full probability distribution of each die and combines them with convolution or by enumerating outcomes, so the results are precise rather than approximate sampled estimates.

How do exploding dice work?

When a die rolls its maximum face, it explodes — you roll it again and add the result, repeating on each new maximum. This calculator caps the explosion depth to keep the math finite, which captures virtually all of the probability mass since deep chains are vanishingly rare.

What is a dice pool success count?

In pool systems you roll many dice and count how many meet or beat a target number. Games like Shadowrun and World of Darkness use this. The tool computes the chance of getting at least a given number of successes using the binomial distribution per die.

How does keep-highest or keep-lowest work?

You roll several dice and keep only the best or worst one — the classic advantage and disadvantage mechanic. The calculator enumerates every combination of dice, applies the keep rule, and tallies the resulting distribution exactly.

Why does the expected value matter?

The expected value is the long-run average result, which helps designers balance mechanics and helps players judge a roll. Combined with the full distribution, it tells you not just the average but how spread out and how swingy a mechanic is.

What is the Dice Tower & Anydice Probability Calculator?

Compute exact probabilities for tabletop dice mechanics — exploding dice, keep-highest or keep-lowest, reroll rules, and threshold success counting for dice pool systems like Shadowrun and World of Darkness. For RPG designers and players. Runs in your browser. It runs free in your browser on Gera Tools, with nothing uploaded.

Dice Tower & Anydice Probability Calculator

Name: Dice Tower & Anydice Probability Calculator
Creator: Gera Tools
License: https://creativecommons.org/licenses/by/4.0/

Designing or playing a system with unusual dice mechanics means knowing the real odds, not guessing. This calculator computes exact probabilities for exploding dice, keep-highest or keep-lowest rolls, and dice-pool success counting, using enumeration and closed-form distributions rather than random simulation.

How it works

Each mechanic uses a precise method:

exploding   : a die at max face rerolls and adds, recursively (depth-capped)
keep N of M : enumerate all M-die outcomes, keep best/worst N, tally the sum
pool success: per die P(hit) = (faces ≥ target)/faces; combine via binomial
expected    : Σ outcome × probability over the full distribution

For exploding and keep mechanics the tool builds the complete probability distribution and reports it as a table. For pool systems it returns the chance of reaching at least your target number of successes, the most common question at the table.

Example and tips

Rolling 4d6 and keeping the highest 3 — the classic ability-score method — has an expected value of about 12.24, not 10.5, because dropping the lowest die shifts the average up. Exploding a d6 raises its mean from 3.5 to 4.2 once chained explosions are included. Use the distribution table to spot swingy mechanics: a high expected value with a wide spread feels very different at the table from a tight one.

Exploding dice: what the distribution actually looks like

An exploding die is interesting because it has no hard maximum. When a d6 rolls a 6, it rolls again and adds — and if that second roll is also a 6, it rolls a third time, and so on. The probability of each chain drops geometrically: a chain of two 6s has probability 1/36, three 6s has 1/216, and so on.

The result is a distribution with a long right tail. Most rolls land in the normal range (1–5 for a d6), but the occasional chain explosion can produce results well above the normal maximum. The calculator caps chain depth to make computation finite, but captures essentially all of the probability mass because deep chains are vanishingly rare.

For a d6:

Mean without exploding: 3.5
Mean with exploding: 4.2 (the extra 0.7 comes entirely from the explosion chains)
The explosion adds modest expected value but substantially increases the maximum possible result

This matters for game design: exploding dice raise the ceiling, not just the floor, which produces memorable outlier moments at the cost of higher variance.

Keep-highest and keep-lowest: advantage and disadvantage

The classic D&D 5e advantage mechanic (roll 2d20, keep highest) is the most familiar version, but the mathematics generalise to any number of dice. The key insight is that keep-highest raises the average result, not by a fixed amount but by an amount that depends on the number of dice rolled.

For 2d20 keep highest:

The probability that the highest of two d20 rolls is 20 is 1 - (19/20)^2 ≈ 9.75% — nearly double the baseline 5%
The expected value rises from 10.5 to approximately 13.8

For 3d20 keep highest (triple advantage, sometimes seen in special circumstances):

The average rises further, to approximately 15.4

Keep-lowest inverts these — it represents disadvantage, making bad outcomes more likely. The distribution table shows the full shape, which is especially useful for system designers deciding how much advantage should shift the odds.

Dice pool success counting for specific systems

In pool systems like Shadowrun, World of Darkness (Classic or Chronicles), and some percentile systems, you roll multiple dice and count successes against a threshold. The key question is always: what is the probability of getting at least X successes from Y dice with a per-die hit chance of P?

This follows a binomial distribution: the probability of exactly k successes from n dice, each with probability p of success, is C(n,k) × p^k × (1-p)^(n-k). The tool sums from your minimum success count upward to give the cumulative probability of meeting or beating your target.

For example, rolling 8 dice with a threshold of 5 on a d6 (needing 5 or 6, so a 1/3 chance per die):

Probability of at least 1 success: about 96%
Probability of at least 3 successes: about 53%
Probability of at least 5 successes: about 12%

These numbers are precise, not simulated — useful both for players evaluating whether to spend a resource, and for designers calibrating how many dice a task should require.