Dice pool calculator
Roll a handful of dice and count every die that reaches the target. Find the chance of scoring at least as many successes as you need, with exploding dice and rerolled ones.
| Successes | Exactly | At least | Odds of at least |
|---|---|---|---|
| 0 | 8.8% | 100% | certain |
| 1 | 26.3% | 91.2% | about 9 in 10 |
| 2 | 32.9% | 64.9% | about 2 in 3 |
| 3 | 21.9% | 32.0% | about 1 in 3 |
| 4 | 8.2% | 10.0% | about 1 in 10 |
| 5 | 1.6% | 1.8% | about 1 in 56 |
| 6 | 0.14% | 0.14% | 1 in 729 |
How to use it
Set how many dice you roll, how many sides they have and the lowest face that counts as a success; 5 on a six-sided die means a 5 or a 6. Then set how many successes you need. The brass plate shows the chance of reaching it, and the table lists every count: exactly is the chance of that many and no more, at least adds up that many or more.
Exploding dice and rerolled ones
With exploding dice, a die that shows the explode face or higher scores as usual and you roll one bonus die, which can explode again, so a pool can beat its own size. Reroll once rolls any 1 again, bonus dice included, and the second result stands.
Each die's chances are worked out exactly, with a short series for the explosions, then combined for the whole pool by convolution. Nothing is simulated.
Sources
- Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, 2nd edition (American Mathematical Society): §7.1 “Sums of Discrete Random Variables” for combining the dice by convolution, and §10.1 “Generating Functions for Discrete Distributions” for the explosion series. Free GNU FDL edition on Dartmouth's site.
Questions
Does a bonus die from an explosion count as a success?
Only if it reaches the target, like any other die. If dice explode on a face below the target, say on 6 with success on 8, that roll adds a bonus die but scores nothing itself.
Why does the table stop with exploding dice?
Explosions can go on and on, so there is no highest count. The table stops once the chance of going further drops below 1 in 100,000, and the at-least column still includes that last sliver.
Why is my chance 0% when I need more successes than dice?
Without explosions every die scores at most one success, so a pool of six dice can never reach seven. Turn on exploding dice if your game allows it.