Yahtzee Probability Calculator
Calculate the exact probability of rolling specific Yahtzee combinations with 5 dice, optionally with up to 2 re-rolls of selected dice.
How many dice you are keeping from a previous roll (0 = fresh start)
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Formulas Used
Single-roll probabilities (5 dice, 65 = 7,776 total outcomes):
- Yahtzee: 6 / 7776 ≈ 0.0772% — 6 ways (one per face)
- Four of a Kind: (6 × C(5,4) × 5 + 6) / 7776 = 156 / 7776 ≈ 2.006%
- Full House: 6 × 5 × C(5,3) / 7776 = 300 / 7776 ≈ 3.858%
- Large Straight: 2 × 5! / 7776 = 240 / 7776 ≈ 3.086%
- Small Straight: |A∪B∪C| / 7776 = 1200 / 7776 ≈ 15.432% (inclusion-exclusion over 3 consecutive-4 sets)
- Three of a Kind: (1200 + 300 + 150 + 6) / 7776 = 1656 / 7776 ≈ 21.296%
Multi-roll (Markov chain): States s = 0…5 matching dice. Transition probability from state s to t (t ≥ s):
P(s → t) = C(5−s, t−s) × (1/6)t−s × (5/6)5−t
State vector is propagated for each roll. Final probability = sum of absorbing-state probabilities.
Compound probability (straights, full house): P(success in r rolls) ≈ 1 − (1 − p₁)r
Assumptions & References
- All dice are fair and independent with faces 1–6 (uniform distribution).
- Optimal strategy is assumed: always keep the maximum number of dice that advance the target combination.
- Single-roll counts verified by exhaustive enumeration of all 7,776 outcomes.
- Multi-roll Yahtzee/4-of-a-kind/3-of-a-kind probabilities use exact Markov chain analysis.
- Multi-roll straight and full house probabilities use the compound formula 1−(1−p₁)ʳ as an approximation (exact values require full state enumeration over partial-straight states).
- Reference: Verhoeff, T. (2001). Optimal Solitaire Yahtzee Strategies. Eindhoven University of Technology.
- Reference: Pawlewicz, J. (2010). Yahtzee: Optimal Strategy. arXiv:1105.0511.
- Standard Yahtzee rules: 5 dice, up to 3 rolls per turn (1 initial + 2 re-rolls).