Farkle Scoring Probability Calculator
Calculate the probability of scoring (avoiding a Farkle) when rolling a given number of dice, plus expected score and risk analysis.
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Formulas & Scoring Rules
Probability of Scoring:
P(score) = (# outcomes with score > 0) / 6n
P(farkle) = 1 − P(score)
E[score] = Σ (scorei × waysi) / 6n
Standard Farkle Scoring:
- Single 1 = 100 pts; Single 5 = 50 pts
- Three of a kind: 1s = 1000, others = face × 100 (e.g. three 4s = 400)
- Four of a kind = triple score × 2; Five = × 4; Six = × 8
- Straight (1–2–3–4–5–6) = 1500 pts
- Three pairs = 1500 pts
- Two triplets = 2500 pts
- Four of a kind + a pair = 1500 pts
All 6n equally-likely outcomes are enumerated via multinomial expansion. Each unique unordered outcome (multiset) has multiplicity n!/(c₁!c₂!…c₆!) where cᵢ is the count of face i.
Assumptions & References
- Standard six-sided fair dice; all faces equally probable (1/6 each).
- Scoring follows widely-used Farkle rules; house rules may vary.
- Special combos (straight, three pairs, two triplets, four+pair) require exactly 6 dice and are checked before standard scoring.
- The calculator enumerates all distinct multisets of n dice rolls exactly — no simulation or approximation.
- Expected score includes farkle outcomes (score = 0); "if scoring" excludes them.
- Recommendation logic: bank if you can win, or if farkle risk > 40% with points at stake.
- Reference: Farkle rules as described by Legendary Games (standard variant).