Examples
Examples
Try the starting example
- Fair die sides
- 6
- Explode on any face at/above this value
- 6
- Maximum EXTRA rolls expanded before unresolved tail
- 2
- At-least total score query
- 20
- Resolved terminal-score probabilities within expansion
- 1: 0.166666666667; 2: 0.166666666667; 3: 0.166666666667; 4: 0.166666666667; 5: 0.166666666667; 7: 0.0277777777778; 8: 0.0277777777778; 9: 0.0277777777778; 10: 0.0277777777778; 11: 0.0277777777778; 13: 0.00462962962963; 14: 0.00462962962963; 15: 0.00462962962963; 16: 0.00462962962963; 17: 0.00462962962963
- Unresolved probability mass beyond expanded rolls
- 0.00463
- Unbounded expected score under stated repeating rule
- 4.2
- Lower bound for queried at-least probability
- 0
- Upper bound for queried at-least probability
- 0.00463
Try another scenario
- Fair die sides
- 6
- Explode on any face at/above this value
- 6
- Maximum EXTRA rolls expanded before unresolved tail
- 0
- At-least total score query
- 20
- Resolved terminal-score probabilities within expansion
- 1: 0.166666666667; 2: 0.166666666667; 3: 0.166666666667; 4: 0.166666666667; 5: 0.166666666667
- Unresolved probability mass beyond expanded rolls
- 0.166667
- Unbounded expected score under stated repeating rule
- 4.2
- Lower bound for queried at-least probability
- 0
- Upper bound for queried at-least probability
- 0.166667
Calculation method
Calculation method
Resolved terminal-score probabilities within expansion
Sum fair-roll sequence counts ending on a nonexploding face, using a common exact denominator.
Unresolved probability mass beyond expanded rolls
Probability every expanded roll explodes; do not silently renormalize resolved scores.
Unbounded expected score under stated repeating rule
E(single fair roll)/(1−explosion probability), requiring at least one nonexploding face.
Lower bound for queried at-least probability
Resolved successes plus unresolved branches already at/above target, since future rolls are positive.
Upper bound for queried at-least probability
Resolved successes plus all unresolved mass; this is a rigorous bound, not a fitted tail.
Symbols and units
- a
- Fair die sides
- b
- Explode on any face at/above this value
- c
- Maximum EXTRA rolls expanded before unresolved tail
- d
- At-least total score query
Use the units shown. A list keeps the order in which you enter its values. The symbol ^ means “raised to the power of”.
Assumptions & limits
Assumptions & limits
- A repeating additive exploding-die model with an explicitly expanded finite prefix and visible unresolved tail. Bounds concern one die chain, not a sum of several chains; no Monte Carlo sampling is used.
Are these inputs private?
These calculations run in your browser. A shared link contains the values you choose to share. Only questions submitted to the separate AI finder are sent to OpenAI.
Further reading
Background reading; this does not imply endorsement or independent review of SumFizz.