Yahtzee Probability & Odds: The Math Behind Every Roll
Yahtzee probability is fully calculable — every roll is just combinatorics. In this guide we derive the exact Yahtzee odds for each category from first principles, so you can stop guessing and start reasoning. Written by a 20-year game systems designer, it uses only the fact that five six-sided dice have 6^5 = 7,776 equally likely outcomes. We also correct a common myth about the "three of a kind" rate you'll see copied across the web.
The Foundation — 7,776 Outcomes
Five independent six-sided dice produce 6 × 6 × 6 × 6 × 6 = 7,776 equally likely results on a single roll. Every probability below is a count of favourable outcomes divided by 7,776. (In a real turn you may reroll up to twice more, so your in-turn rates are higher than these single-roll figures — more on that at the end.)
Exact Category Breakdown (Mutually Exclusive)
To avoid double-counting, we split rolls into non-overlapping patterns. The counts sum to exactly 7,776.
| Exact pattern | Count | Math |
|---|---|---|
| 5 of a kind (Yahtzee) | 6 | 6 faces |
| Exactly 4 of a kind | 150 | 6 × 5 × C(5,4) |
| Exactly full house (3+2) | 300 | 6 × 5 × C(5,3) |
| Exactly 3 of a kind | 1,200 | 6 × C(5,3) × 5 × 4 |
| Exactly two pairs | 1,800 | C(6,2) × C(5,2) × C(3,2) × 4 |
| Exactly one pair | 3,600 | 6 × C(5,2) × 5 × 4 × 3 |
| All five distinct | 720 | 6 × 5 × 4 × 3 × 2 |
| Total | 7,776 | — |
First-Roll Probabilities (What the Scorecard Cares About)
| Outcome (at least) | Count | Probability |
|---|---|---|
| Yahtzee (5 of a kind) | 6 | 6 / 7,776 ≈ 0.08% (1 in 1,296) |
| Four of a kind | 6 + 150 = 156 | 156 / 7,776 ≈ 2.01% |
| Full House | 300 | 300 / 7,776 ≈ 3.86% |
| Large Straight | 240 | 240 / 7,776 ≈ 3.09% |
| Small Straight* | 1,800 | 1,800 / 7,776 ≈ 23.15% |
| Three of a Kind (incl. 4 & 5) | 6 + 150 + 300 + 1,200 = 1,656 | 1,656 / 7,776 ≈ 21.30% |
Myth correction: You will see "Three of a Kind ≈ 44.44%" (3,456 / 7,776) copied across the web. That count actually equals three-of-a-kind OR two-pairs (1,656 + 1,800), not three-of-a-kind alone. The true "at least three of a kind" rate is 21.30%.
Why This Matters for Strategy
- Yahtzee is rare (0.08% per roll). Treat it as a bonus, never a plan — confirmed by the math, not folklore.
- Straights and three-of-a-kinds are common (23% and 21%). You can safely build a strategy around them.
- The upper bonus is the lever. Because most patterns are "at least" events, banking guaranteed upper-section points beats gambling for the elusive 50.
Expected Value — A Worked Example
Roll 4-4-4-2-6 on your last reroll with the Yahtzee box closed: Four of a Kind = sum = 20. Safe, above average. Fours (upper) = 12, pushing toward the +35 bonus. Yahtzee attempt: reroll the 2 and 6; you need both to become 4s — probability (1/6)² ≈ 2.8%. Not worth it. The EV-maximising play is usually the sure upper points, not the 2.8% lottery.
Within a Turn (Rerolls Raise the Rates)
A turn allows up to three rolls. The single-roll figures above are floors. The well-established result: the chance of completing a Yahtzee within a full turn (optimal play) is about 4.6%, and straights/sets rise correspondingly. We won't derive the full Markov chain here, but the principle holds — rerolls convert rare single-roll events into workable in-turn odds.