Odds of rolling a Yahtzee
That makes Yahtzee the 2nd lowest-scoring category of 13 on the sheet, if you spend a whole turn chasing it and nothing else. It pays a fixed 50 when you get it, so the expected score is just that times the completion chance.
You rolled a second Yahtzee. What now?
If your Yahtzee box already shows 50, every further Yahtzee is worth 100 bonus points, with no limit on how many you can score. If you took a zero in the Yahtzee box earlier, you get nothing — the bonus is forfeit for the rest of the game. That is the half people miss.
You still have to write the roll down somewhere, and the official rule does not let you choose freely. If the upper box for that face is open, it must go there. If it is already filled, the roll becomes a joker in any open lower box — and Full House, Small Straight and Large Straight pay their full 25, 30 and 40 even though the dice are not that shape. Only if the lower section is full too do you take a zero in an open upper box.
Plenty of households play a looser version where you pick any open box. Neither is wrong, but the two score differently, so settle it before the first roll rather than after this one. The full-game solver assumes the looser convention, and says so.
What the rerolls buy you
A turn is three rolls: the first, then two rerolls of any dice you don’t keep. Chasing Yahtzee from the first roll alone is worth 0.04; using the full turn is worth 2.30.
| Rolls available | Expected score | Chance of hitting it |
|---|---|---|
| First roll only | 0.04 | 0.08% |
| First roll + 1 reroll | 0.63 | 1.26% |
| Full turn (2 rerolls) | 2.30 | 4.60% |
Each row assumes you reroll optimally for this category alone, and nothing else. The two rerolls multiply the first-roll figure by 59.6×.
Every score you can end up with
Expected score is an average, and averages hide the shape. Yahtzee only ever ends one of two ways: you get it, or you don’t. You end the turn with nothing 95.40% of the time.
Exact probability of each final score, playing the whole turn for Yahtzee. The highlighted bar is the likeliest outcome, not the best one.
How much the first roll decides
Once the first roll lands, you are in one of only 5 genuinely different positions for Yahtzee — every roll inside a tier is worth exactly the same, because the optimal reroll treats them identically.
| Example roll | How often | Turn is now worth |
|---|---|---|
| 0.08% | 50.00 | |
| 1.93% | 15.28 | |
| 19.29% | 4.67 | |
| worst tier | 9.26% | 0.63 |
Top three tiers, then the worst one. The worst tier covers 6 of the 252 distinct opening rolls and lands 9.26% of the time.