Yahtzee Ones odds
That makes Ones the 1st lowest-scoring category of 13 on the sheet, if you spend a whole turn chasing it and nothing else.
What the rerolls buy you
A turn is three rolls: the first, then two rerolls of any dice you don’t keep. Chasing Ones from the first roll alone is worth 0.83; using the full turn is worth 2.11.
| Rolls available | Expected score | Chance of scoring |
|---|---|---|
| First roll only | 0.83 | 59.81% |
| First roll + 1 reroll | 1.53 | 83.85% |
| Full turn (2 rerolls) | 2.11 | 93.51% |
Each row assumes you reroll optimally for this category alone, and nothing else. The two rerolls multiply the first-roll figure by 2.5×. That last column is the same for all six upper categories, and so is that multiplier: a die is equally likely to show a 1 as a 6. The faces differ in what they pay, never in how often they turn up.
Every score you can end up with
Expected score is an average, and averages hide the shape. Playing a full turn for Ones, the single likeliest result is 2 at 34.40%, and you come away with nothing 6.49% of the time.
Exact probability of each final score, playing the whole turn for Ones. 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 6 genuinely different positions for Ones — 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 |
|---|---|---|
| 0.01% | 5.00 | |
| 0.32% | 4.31 | |
| 3.21% | 3.61 | |
| worst tier | 40.19% | 1.53 |
Top three tiers, then the worst one. The worst tier covers 126 of the 252 distinct opening rolls and lands 40.19% of the time.
Ones and the upper bonus
The upper section pays 35 bonus points at 63, which is exactly three of every face. So Ones has to carry 3 to hold that pace. Chasing it for a whole turn averages 2.11 — short of pace by 0.89, which has to come back from a higher face. This is why the bonus is usually won or lost on fives and sixes.