Mah Jongg odds: a cheat sheet
The chances worth knowing at an American Mah Jongg table: what the set and the deal hold, and how likely you are to draw the tiles you need.
These are the odds before anything tells you where the tiles are. Once the Charleston, the discards and the exposures show you tiles, fewer are left unseen and the odds change. The hints at the table count what you've seen.
The set and the deal
| Tiles | Kinds | Of each | In all |
|---|---|---|---|
| Bams, Craks and Dots, 1 to 9 | 27 | 4 | 108 |
| Winds | 4 | 4 | 16 |
| Dragons | 3 | 4 | 12 |
| Flowers | 1 | 8 | 8 |
| Jokers | 1 | 8 | 8 |
| The set | 152 |
East is dealt 14 tiles and the others 13, which leaves 99 in the wall. If no one calls a discard, each player draws 24 or 25 tiles before the wall runs out. A call skips the players in between, so when someone calls, some players draw fewer.
Jokers in your deal
There are 8 jokers in the set. How many a deal holds:
| Jokers | Your 13 tiles | East's 14 |
|---|---|---|
| none | 48.0% | 45.3% |
| one | 37.9% | 38.7% |
| two | 12.0% | 13.3% |
| three | 2.0% | 2.4% |
| four or more | 0.2% | 0.3% |
On average a deal holds 0.68 jokers: 48.0% of deals hold none, and 14.1% hold two or more. The set has as many flowers as jokers, so the same goes for the flowers in a deal.
Drawing what you need
Your outs are the tiles that would help: the copies of a tile you need that you haven't seen, and for a group of 3 or more, the jokers too. When play begins you've seen your own 13 tiles, so 139 are unseen. The chance that at least one of your outs comes in your next draws:
| Outs | Next draw | Next 5 draws | Next 10 draws | Next 24 draws, about all you get |
|---|---|---|---|---|
| 1 | 0.7% | 3.6% | 7.2% | 17.3% |
| 2 | 1.4% | 7.1% | 13.9% | 31.7% |
| 3 | 2.2% | 10.5% | 20.2% | 43.6% |
| 4 | 2.9% | 13.8% | 26.1% | 53.6% |
| 6 | 4.3% | 20.1% | 36.7% | 68.7% |
| 8 | 5.8% | 26.0% | 45.9% | 79.0% |
| 10 | 7.2% | 31.5% | 53.8% | 86.0% |
| 12 | 8.6% | 36.8% | 60.7% | 90.7% |
Each draw on its own is a small chance: your outs out of the tiles unseen. Later in a game you've seen more tiles, so each draw's chance is a little better, but there are fewer draws left.
Why jokers matter
Say you hold two 5 Dots, and no other 5 Dot has been seen. To make them a pung, either of the other 2 or any of the 8 jokers will do: 10 outs. To make a pair of one 5 Dot, only its own tile will do: 3 outs. Over your next 10 draws, that's 53.8% for the pung against 20.2% for the pair.
That's the joker rule at work: a joker stands in only within a group of 3 or more, never in a single or a pair. So a pair's last tile is the hardest kind to draw, and a hand that needs it is a long way off.
Jokers and winning
How often a player wins, by the jokers dealt them, when all four players play equally well:
| Jokers dealt | Wins |
|---|---|
| none | 13.3% |
| one | 23.5% |
| two | 35.3% |
| three or more | 55.8% |
A deal with three or more jokers wins about 4 times as often as one with none. In a single game the deal decides a lot, so the Game Review says when it did. This is the one number here that's measured, not worked out (below).
Where these numbers come from
The app works out these numbers itself, by counting. For a deal, it counts every way the tiles can be dealt from the set. For your draws, it counts every way your next draws can come from the tiles you haven't seen. From your seat, each of those is as likely as any other to be among your draws, since the wall's order and the other hands are hidden alike.
The one measured number is how often a player wins by their jokers: from robot games, all four at the same level, on the app's practice card.
How the app's engine counts with them, at the table and in the Game Review: How it works.