Math

Determining the probability to win Majong, an unfair game?

Introduction

Mahjong is a traditional Chinese tile-based game commonly played between four players. Mahjong involves skill, strategy, calculation and most importantly a degree of chance.

The higher Fann value (points for winning the game) is, the less likely it will appear in the game. The aim of this exploration is to investigate the probability of winning and use it to determine the fairness of the scoring system. I will establish a general probability calculation model and make assumptions to calculate the probability of various type of combination to win based on four ways of winning, namely, the Big Three Dragons, Small Three Dragons, Junior Winds Hand, and Pure Hand.

Rationale

Mahjong and other forms of gambling games all have slim probability to win yet people are still drawn to it due to its seemingly grand price. I want to explore the reward from this gambling game as it is a part of our cultural that’s worth looking into. I grow up in a mahjong family and want to take so mahjong holds a significant importance to me.

Introduction to Majong

Basic Terminology in Majong

There are 3 suits of simples (dots, bamboo, and characters) with nine tiles in each labelled from 1 to 9. There are 4 copies of each simples tile totaling 108 simples tiles. In addition, there are dragons in the color red, green, and white and winds (south, east, west, and north).

Seasons and flower is not used in the IA (exclusive) Total tiles: 136

Formations in Majong

· Pong: a set of three identical tiles, for example 3 red dragons

· Kong: a set of four identical tiles, for example, 4 red dragons

· Chow: 3 consecutive tiles in a simple, for example, bamboo labelled 1, 2, and 3

· Pair/eye: a set of two identical tiles, for example, two red dragons

Ways to win in Majong (Fann value in descending order)

· Big Three Dragons: consists of all the dragon tiles, one pair of pong or chow, and an eye

· Small Three Dragons: has two types of dragons tiles plus two melds/chows of any tiles and an eye

· Junior winds hand: has three melds and a pair of winds plus one meld/chow of any tiles

· Pure hand: has one suit, for example, 1 to 9 for bamboo simples.

The player wins from collecting the tiles either from the wall or from the tiles other player discarded.

Exploring the probability to win

There will be a total of 53 tiles among the four players by the end of each round, which means there are kinds of combinations. (winning hand has 14 tiles; other three players have 13 tiles)

Assumptions:

1. The competition is relatively fair, participants at a comparable level and perform normally.

2. Do not use seasons and flowers and the total number of a set of Mahjong is 136.

3. Drawing and discarding are tile are random events, and the probabilities of the appearance of discarded tiles and undrawn tiles are equal.

4. The probability of a hand to form meld/chow/kong/eyes is directly proportional to the probability of a set of Mahjong to form meld/chow/kong/eyes randomly.

5. The probability of a type of winning hand is constant

Exploring the probability of winning

1. Big Three Dragons

Since the probabilities of the appearance of discarded tiles and undrawn tiles are equal, so the probability of nine dragons in one-person hand is


53 - 9 = 44 tiles.

136 - 9 = 127 tiles

After picking the three dragons away, there are kinds of combination left.

Therefore, the probability of three dragon in one-person hand and win the game is

Chow ABC: It is known from the assumption 4 that the probability of the random composition of chow ABC:

*7: There are 7 types of chows’ ABC in each suit, which is

123,234,345,456,567,678,789.

Pong AAA: the probability of the random group of AAA as Dots/Bamboo/Characters/Winds:

.

Eyes AA: the probability of the random composition of Eyes AA as Dots/Bamboo/Characters/Winds is:

Therefore, the equation for a and b is:

a.

b.

The probability of the two types of way to combine Bing Three Dragon is equal, so we have

Hence from the analysis above, the probability of Big Three Dragon is:

Reflection:

This result shows the probability of Big Three Dragons is quite rare. The reward is 88 fann (for example, put 100 dollars in to the game you will get 8400 dollars at the end), which is not fair comparing to the probability we calculate above.

2. Small three dragons

The Small three dragon includes two group of three dragons so the combination of it is

Since the probabilities of the appearance of discarded tiles and undrawn tiles are equal, so the probability of eight dragons consist in one hand is

As for the other 47 tiles, there are kinds of combinations.

Three types of winning hands of Junior dragons hand:

a. Dragons*3+Dragons*3+ABC+ABC+aa(Dragons)

b. Dragons*3+Dragons*3+ABC+AAA+aa(Dragons)

c. Dragons*3+Dragons*3+AAA+AAA+aa(Dragons)

It is known from the assumption 4 that the probability of the random composition of chow ABC

is:

The probability of the random composition of Meld AAA as Dots/Bamboo/Characters/Winds is:

The probability of the random composition of Eyes aa as Dragons is:

As for the Eyes and 2 Chows in winning hand type 1, the probability is:

As for the Eyes, Meld and Chow in winning hand type 2, the probability is:

As for the Eyes and 2 Melds in winning hand type 3, the probability is:

From the above analysis, the probability of two melds and a pair of eyes of dragons consist in one hand and win is

According to assumption 5, the probabilities of the above three types of wining hand are equal, is

Hence, the probability of Small three dragons is:

Reflection:

Although the probability of small three dragons is much more likely to happen compare to big three dragons but compare to the the fann value (64 fann), it is still unfair.

3. Junior winds hand

‘Junior winds hand’ consists three melds of winds, so the number of combination of them is

Since the probabilities of the appearance of discarded tiles and undrawn tiles are equal, so the probability of nine winds consist in one hand is

As for the other 44 tiles, there are kinds of combinations.

Therefore, the equation of Junior winds had is:

a. Winds*3+Winds*3+Winds*3+ABC+aa(Winds)

b. Winds*3+Winds*3+Winds*3+AAA+aa(Winds)

Chow ABC: it is known from the assumption 4 that the probability of the random composition of chow ABC is:

Meld AAA: the probability of the random composition of Meld AAA as Dots/Bamboo/Characters/Dragons is:

Eyes aa: the probability of the random composition of Eyes aa as Winds is:

According to assumption 6, the probability of a hand to form the above three analyses are:

i.

ii.

The probability of all three wind title in one person hand and win is:

According to assumption 5, the probabilities of the above two types of wining hand are equal, is

Thus, the possibility of Junior winds hand is:

Reflection:

Junior wind hand has the same value with small three dragons. (64 fann). However, by looking at the calculation of these two equation, it shows that junior wind hand is more likely to appear compare to Small three dragons, which reflected that the original scoring criteria is unfair.

4. Pure hand

‘Pure hand’ consists fourteen tiles in the same suit, so the number of combination of ‘Pure hand’ is

As for the other 39 tiles in the rest hands, there are kind of way to combine.

There are 5 ways to combine pure hand:

a. aa+ABC+ABC+ABC+ABC

b. aa+ABC+ABC+ABC+AAA

c. aa+ABC+ABC+AAA+AAA

d. aa+ABC+AAA+AAA+AAA

e. aa+AAA+AAA+AAA+AAA

Chow ABC in any suit: it is known from the assumption 4 that the probability of the random combination of chow ABC is:

Meld AAA: the probability of the random composition of Meld AAA as the same suit with Chow ABC is:

Eyes aa: the probability of the random composition of Eyes aa as the same suit with Chow ABC is:

Basic on the calculation above, the possibility of the 5 types of combination is (according to assumption 6):

a.

b.

e.

d.

e.

The probability of 14 tiles in the same suit in one-person hand is:

According to assumption 5, the probabilities of the above five types of wining hand are equal. So, the possibility is

Hence, the equation of pure hand is:

Reflection:

Pure hand valued the less fann (24 fann), however by looking at the probability (), it is overvalued, because it is more likely to appear compare to the other three ways.

Overall Reflection and Conclusion:

Based on the results of my finding, I discovered that the scoring system (Fann value) is unfair as the probability for winning is low and the prize obtained is not very high. Similar to all the other gambling games, the prize and the payment for entering the game is often unequal and the odd is often not on the player’s side.

Overall, my exploration fail to explore all the ways to win in Majong. However, to make the evaluation efficient, I chose four winning ways with different fann values – a highest fann, a lowest fann, and other two identical values in the middle to discover the general trend in majong and understand the underlying mathematic behind this game. There are many ways to win in Mahjong as it is a very complicated game, however due to the limited space, I only a calculated 4 type of winning hand. I still can’t tell if the scoring criteria is fair or not.

Bibliography

Mahjong Competition Rules . World Mahjong Organization, 2005, mahjong-europe.org/portal/images/docs/mcr_ENa5.pdf.

“Mahjong Wiki (麻将维基).” Suits - Mahjong Wiki (麻将维基), mahjong.wikidot.com/big-three-dragons.

“The Rules of Mah Jong (Chinese & British Versions).” The Rules of Mahjong, www.mastersofgames.com/rules/mah-jong-rules.htm.