How long until my hand improves?
A practical way to turn ukeire into expected draws, with a reference table and a closer look at the assumptions.
From ukeire to a waiting time
How long should we expect to wait when a hand has T useful tiles among N unknown tiles? This estimate covers hands through 1-shanten and ignores ron, calls, and opponent interaction.
The quick estimate is N / T. A better without-replacement estimate for the first useful draw is (N + 1) / (T + 1).
At the beginning of a hand, 14 tiles are in our hand and one dora indicator is visible, leaving about N = 121 unknown tiles. After five, ten, and fifteen draws by each player, use approximately 100, 80, and 60 unknown tiles. A full hand contains 70 self-draws, though rounds usually end earlier.
Expected draws by ukeire
Useful tiles T |
121 left | 100 left | 80 left | 60 left |
|---|---|---|---|---|
| 28 | 4.2 | 3.5 | 2.8 | 2.1 |
| 24 | 4.9 | 4.0 | 3.2 | 2.4 |
| 20 | 5.8 | 4.8 | 3.9 | 2.9 |
| 16 | 7.2 | 5.9 | 4.8 | 3.6 |
| 12 | 9.4 | 7.8 | 6.2 | 4.7 |
| 9 | 12.2 | 10.1 | 8.1 | 6.1 |
As the wall gets shorter, the same ukeire becomes more urgent.
A worked example
Ryo Hanyu's analysis considers a non-tenpai hand with 18 useful tiles. Starting from 121 unknown tiles, the chance of at least one improvement in the next k draws is 1 − C(121 − 18, k) / C(121, k): about 14.9% after one draw, 27.7% after two, and 61.8% after six. After the first row, roughly 24 tiles are visible and the six-draw chance rises to about 67.5%.
Use these figures to compare shapes quickly. A tile can improve shanten while making the eventual wait weak, and a live tile may still be valuable for defence or a high-value route.