Yatzy Odds: Probability of a Yatzy and a Large Straight

2026-10-30

Here are the Yatzy odds, worked out exactly: five alike on the first roll of a turn comes up 6 times in 7,776 throws, which is 1 in 1,296 (0.08%), and spending all three rolls chasing it lifts that to 4.60%, about 1 in 22. A large straight is 3.09% on the first roll and a full house 3.86%. Every odds figure in the tables is exact: counted over all 7,776 throws of five dice, or calculated over the three rolls of a turn.

Yatzy odds on the first roll

Five dice have 6 × 6 × 6 × 6 × 6 = 7,776 equally likely outcomes. Counting how many of them make each combination gives this table. "At least" means the combination is present, so four alike includes five alike.

Full House is the strict three-plus-two, so five alike does not count. Two Pairs needs two different faces, so four alike does not count either.

How the Yatzy and large-straight numbers are worked out

Probability of a Yatzy. Choose which of the six faces it will be, then all five dice must land on it, each with probability 1/6: 6 × 1/6 × 1/6 × 1/6 × 1/6 × 1/6 = 6/7,776 = 1/1,296.

Probability of a large straight. There are two runs, 1-2-3-4-5 and 2-3-4-5-6. Five different faces can fall in 5! = 120 orders, so 2 × 120 = 240 throws out of 7,776, or 3.09%. On the Scandinavian sheet one specific run counts per box, so each box gets half of that: 120 throws, 1.54%.

Odds within three rolls

Throw, keep what helps, re-roll the rest, twice more. This table gives the best chance of ending the turn with a combination when it is the only thing you want and you hold the dice perfectly.

Read these as ceilings. A real turn has several boxes open and you rarely give all three rolls to a single target, so the chance of actually landing any one combination is lower than its row. They still work as a price list: a Classic small straight arrives about three times in five when you go for it, a large straight about one time in four, a Yatzy about one time in twenty-two.

Chasing a Yatzy: what each held group is worth

The most common decision in the game is how many dice to keep. This table shows the chance of finishing with five alike when you hold a group of one face and re-roll everything else.

With one roll left every missing die has to match, so each one multiplies the odds by 1/6: for four alike down to one die that is 1/6, 1/36, 1/216 and 1/1,296, the 16.7%, 2.8%, 0.5% and 0.1% of the first column. With two rolls left and four alike in hand you only fail by missing twice: 1 − (5/6) × (5/6) = 1 − 25/36 = 11/36, which is 30.6%.

Play bears this out. Over 800 bot-against-bot Classic games per level, the hard bot, which calculates every roll left in the turn, scores the Yatzy box in about 36% of games and writes a zero there in about 64%.

Odds of getting two Yatzys in a match

There is no clean figure, and we will not invent one. A second Yatzy needs the same long shot to come off again, and how often you even try depends on how you play the other turns, so only a simulation of a stated strategy could give a number. What the rules do fix is the payoff. On the Classic sheet an extra Yatzy adds 100 points, but only if the Yatzy box already holds 50; a scratched zero pays nothing extra. The Scandinavian and Yacht sheets have no extra-Yatzy bonus. The full rules are in the Yatzy rules guide.

The upper bonus: why 63

Three of every face gives 3 × (1 + 2 + 3 + 4 + 5 + 6) = 63, exactly the bonus threshold. Every die short of three of a face costs you that face value; every die above three earns it back. The bonus is +35 on Classic and Yacht, +50 on Scandinavian, and the Generala sheet has none. Measured bot against bot, the hard bot reaches 63 in roughly 43% of Classic games and roughly 70% of Scandinavian ones; the medium bot in about 16% and 49%.

Generala: served odds

On the Generala sheet a combination made on the first roll of the turn ("servida") scores 5 extra on Escalera, Full and Póker: 25, 35 and 45. These are the first-roll figures from the table above:

• Served Generala (five alike): 6 of 7,776, 1 in 1,296. It scores 50 whenever it arrives and gets no +5.
• Served Escalera: 360 of 7,776, 4.63%. The two runs 1-2-3-4-5 and 2-3-4-5-6 make 240 throws, and the ace-high 3-4-5-6-1, which Escalera also accepts, adds another 120.
• Served Full: 300 of 7,776, 3.86%.
• Served Póker: 156 of 7,776, 2.01%, four alike or better.

Some house rules make a served Generala an instant win. Here it does not: it writes 50 like any other. The Generala rules cover that sheet; the Yacht rules cover the sheet with 15 and 30 point straights.

The dice are fair, and you can check

The dice of a match are sealed before it starts. The server publishes only a fingerprint, a SHA-256 hash of a secret seed; each human player adds a random seed from their browser before their first roll; every roll is derived from the two. Bytes that would favour a face are discarded, so 1 to 6 are exactly equally likely, and no table theme can touch a die. When the match ends the seed is revealed, and "Verify the rolls" recomputes every roll in your own browser. The odds in this article are the odds at the table.

For scale, perfect play on the American 13-box sheet, which uses the Classic rules here, averages about 254.6 points with a standard deviation near 60, according to the optimal solitaire player built by T. Verhoeff (TU Eindhoven, 1999); our hard bot averages about 241 on Classic. Use the tables as a price list when you choose what to keep, read the Yatzy strategy tips for the decisions behind them, and test them in a free match at Yatzy online.

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