A Python package for computing dice probabilities.
- Pure Python implementation using only the Standard Library. Run it anywhere Python runs: program locally, share Jupyter notebooks, or build your own client-side web apps using Pyodide.
- Dice support all standard operators (+, -, <, >, etc.) as well as an extensive library of functions (rerolling, exploding, etc.)
- Efficient dice pool algorithm can solve keep-highest, finding sets and/or straights, RISK-like mechanics, and more in milliseconds, even for large pools.
- Exact fractional probabilities using Python
ints. - Experimental support for decks (aka sampling without replacement).
pip install icepool
The source is pure Python, so including a direct copy in your project can work as well.
Feel free to open a discussion or issue on GitHub. You can also find me on Twitter or Reddit.
See this JupyterLite distribution for a collection of interactive, editable examples. These include mechanics from published games, StackExchange, Reddit, and academic papers.
These are all client-side, powered by Pyodide. Perhaps you can use them as inspiration for your own application.
- Icecup, a simple frontend for scripting and graphing.
- Ability score rolling method calculator.
- Cortex Prime calculator.
- Legends of the Wulin calculator.
Published in Artificial Intelligence and Interactive Digital Entertainment (AIIDE) 2022.
BibTeX:
@inproceedings{liu2022icepool,
title={Icepool: Efficient Computation of Dice Pool Probabilities},
author={Albert Julius Liu},
booktitle={Eighteenth AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment},
volume={18},
number={1},
pages={258-265},
year={2022},
month={Oct.},
eventdate={2022-10-24/2022-10-28},
venue={Pomona, California},
url={https://ojs.aaai.org/index.php/AIIDE/article/view/21971},
doi={10.1609/aiide.v18i1.21971}
}What's the chance that the sum of player A's six ability scores is greater than or equal to the sum of player B's six ability scores? (Using 4d6 keep highest 3 for each ability score.)
import icepool
single_ability = icepool.d6.keep_highest(4, 3)
# The @ operator means: compute the left side, and then roll the right side that many times and sum.
print(6 @ single_ability >= 6 @ single_ability)Denominator: 22452257707354557240087211123792674816
| Outcome | Weight | Probability |
|---|---|---|
| False | 10773601417436608285167797336637018642 | 47.984490% |
| True | 11678656289917948954919413787155656174 | 52.015510% |
Roll a bunch of dice, and find all matching sets (pairs, triples, etc.)
We could manually enumerate every case as per the blog post. However, this is prone to error. Fortunately, Icepool can do this simply and reasonably efficiently with no explicit combinatorics on the user's part.
import icepool
class AllMatchingSets(icepool.OutcomeCountEvaluator):
def next_state(self, state, outcome, count):
"""next_state computes a "running total"
given one outcome at a time and how many dice rolled that outcome.
"""
if state is None:
state = ()
# If at least a pair, append the size of the matching set.
if count >= 2:
state += (count,)
# Prioritize larger sets.
return tuple(sorted(state, reverse=True))
all_matching_sets = AllMatchingSets()
# Evaluate on 10d10.
print(all_matching_sets(icepool.d10.pool(10)))Die with denominator 10000000000
| Outcome | Quantity | Probability |
|---|---|---|
| () | 3628800 | 0.036288% |
| (2,) | 163296000 | 1.632960% |
| (2, 2) | 1143072000 | 11.430720% |
| (2, 2, 2) | 1905120000 | 19.051200% |
| (2, 2, 2, 2) | 714420000 | 7.144200% |
| (2, 2, 2, 2, 2) | 28576800 | 0.285768% |
| (3,) | 217728000 | 2.177280% |
| (3, 2) | 1524096000 | 15.240960% |
| (3, 2, 2) | 1905120000 | 19.051200% |
| (3, 2, 2, 2) | 381024000 | 3.810240% |
| (3, 3) | 317520000 | 3.175200% |
| (3, 3, 2) | 381024000 | 3.810240% |
| (3, 3, 2, 2) | 31752000 | 0.317520% |
| (3, 3, 3) | 14112000 | 0.141120% |
| (4,) | 127008000 | 1.270080% |
| (4, 2) | 476280000 | 4.762800% |
| (4, 2, 2) | 285768000 | 2.857680% |
| (4, 2, 2, 2) | 15876000 | 0.158760% |
| (4, 3) | 127008000 | 1.270080% |
| (4, 3, 2) | 63504000 | 0.635040% |
| (4, 3, 3) | 1512000 | 0.015120% |
| (4, 4) | 7938000 | 0.079380% |
| (4, 4, 2) | 1134000 | 0.011340% |
| (5,) | 38102400 | 0.381024% |
| (5, 2) | 76204800 | 0.762048% |
| (5, 2, 2) | 19051200 | 0.190512% |
| (5, 3) | 12700800 | 0.127008% |
| (5, 3, 2) | 1814400 | 0.018144% |
| (5, 4) | 907200 | 0.009072% |
| (5, 5) | 11340 | 0.000113% |
| (6,) | 6350400 | 0.063504% |
| (6, 2) | 6350400 | 0.063504% |
| (6, 2, 2) | 453600 | 0.004536% |
| (6, 3) | 604800 | 0.006048% |
| (6, 4) | 18900 | 0.000189% |
| (7,) | 604800 | 0.006048% |
| (7, 2) | 259200 | 0.002592% |
| (7, 3) | 10800 | 0.000108% |
| (8,) | 32400 | 0.000324% |
| (8, 2) | 4050 | 0.000041% |
| (9,) | 900 | 0.000009% |
| (10,) | 10 | 0.000000% |
In roughly chronological order:
http://hjemmesider.diku.dk/~torbenm/Troll/
The oldest general-purpose dice probability calculator I know of. It has an accompanying peer-reviewed paper.
Probably the most popular dice probability calculator in existence, and with good reason---its accessibility and shareability remains unparalleled. I still use it often for prototyping and as a second opinion.
SnakeEyes demonstrated the viability of browser-based, client-side dice calculation, as well as introducing me to Chart.js.
https://gist.github.com/vyznev/8f5e62c91ce4d8ca7841974c87271e2f
This demonstrated the trick of iterating "vertically" over the outcomes of dice in a dice pool, rather than "horizontally" through the dice---one of the insights into creating a much faster dice pool algorithm.
https://github.com/posita/dyce
Another Python dice probability package. I've benefited greatly from exchanging our experiences.