About Mahjong Probability
Mahjong is a very complex game involving too many variables to calculate an accurate chance of winning. For example, a player can draw a strong starting hand, and that in itself is hard to calculate. What are the chances of drawing good tiles and what are the chances of that affecting the winning outcome?
Of course, adding in other variables like the ability of the player to discard a tile strategically, there can be a countless amount of possibilities affecting the winning hand, which can perhaps be modelled on a program to get a viable range of answers.
In most cases, the answers are not particularly useful in deciding if “I will win this game of Mahjong”, but for a straightforward calculation, here is an attempt.
Mahjong Winning Probability
What are the chances of winning Mahjong
As four players are in the game, a 25% chance. However, to test it out and observe how that might change, here are the chances for a game played for 100 rounds between 4 players of a mahjong game. (NOTE: This is a very general/highly simplified representation based on the game run in a script code.)
1st 100 rounds: [0.28, 0.24, 0.22, 0.23]
2nd 100 rounds: [0.19, 0.21, 0.27, 0.18]
3nd 100 rounds: [0.22, 0.24, 0.2, 0.22]
The above assumes each player as being independent of the other player, which explains why the probabilities do not add up to a 100%. The 4 digits each represent, Player 1, Player 2, Player 3 and Player 4 respectively.
In this case, it seems that player 1 was very lucky in round 1 with a 28% chance of winning, while the lowest chance of winning stood around 18%. To take this further would be to calculate the statistical significance, test the accuracy of the calculations and so on.. But as a normal person, 25% would be a good enough estimate!
What are the chances of a specific winning hand?
As a general representation of a winning hand let's take a look at the chances of an "All Chows" (平和) winning condition in Mahjong. Here are the chances for a game played for 100 rounds between 4 players of a mahjong game.
1st 100 rounds: [0.1, 0.11, 0.07, 0.03]
2nd 100 rounds: [0.05, 0.06, 0.1, 0.04]
3nd 100 rounds:[0.09, 0.07, 0.09, 0.07]
Falling anywhere between the 5% to 10% range, that is the likely chance of an “All Chows” in a mahjong game.
More on the probability of an all Chow in code: However this too is not the accurate measurement, the calculations are run based on the assumption that:
chow_tiles = ['Character', 'Bamboo', 'Dot']
Explanation: These are the three suits in Mahjong that can form sequences (chows).
bonus_tiles = ['Flower', 'Season']
In Mahjong, Flower and Season are bonus tiles that are not part of the main hand and don't contribute to the sequence (chow) combinations.
if len(hand) != 14: return False
A winning Mahjong hand typically consists of 14 tiles.
if all(tile in chow_tiles for tile in hand) and not any(tile in bonus_tiles for tile in hand):
return True
This checks that every tile in the hand belongs to the chow tiles (i.e., Character, Bamboo, or Dot), and that no bonus_tiles are present. However, for this last condition, though the logic was correct it was not entirely complete.
Chows are Specific Sequences: A key part of the "All Chows" condition is that the hand must be made up of sequences of three consecutive tiles (e.g., 1-2-3 of Bamboo). The code only checks if the hand consists of valid tiles from the three suits, but it does not check if the tiles form valid chows (sequences). The current code would consider a hand of 3 characters to be valid, even though it isn't a sequence.
Therefore, the probabilities are again a general measure of how likely it is to get an All Chow(平和) !
So is Mahjong a game of luck?
Mostly, yes, a game of luck and skill. With a ratio of skills to luck at 7:3.(More on luck and technique here). With nothing in the game being equal or statistically calculable (each individual displays a different playstyle and skill level added on to the randomness of luck) there will never be a “correct” answer.
More importantly, enjoy the game of mahjong! There are days where beginner's luck may fall on the hands of the first-time players or you may feel particularly lucky. Pair that with Customymahjong’s custom mahjong tiles and play your way to victory!
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