Mahjong

Because of optimistically economic expectation, I recently received many wedding banquet invitations. In a Chinese wedding banquet, Mahjong game (Chinese: 麻將, Japanese: 麻雀 [まあじゃん] ) is a must. Chinese guests like to play Mahjong before a banquet starts.

For the history and rules of Mahjong, you can refer to Wikipeda (English, Chinese, Japanese). I am just a beginner and know a little about the game. But I find out the following interesting figures.

For simplifying the calculation, the following assumptions are made:

1. No Flower Tile (Chinese: 花牌) is involved. So the total number of tiles is 136. After assigning 13 tiles to each player, there are 84 tiles remained on the table.
2. You are the first player.
3. Your opponents do not keep or later take your targeted tiles.
4. No one Pong (Chinese: 碰), Chi (Chinese: 吃, Cantonese: 上 [Sheung] ), Kong (Chinese: 槓) before you complete the targeted pattern.

Case 1a: If you want to have Three Identical Tiles or Pong but you have none of targeted tile. Fortunately, you can take each of them in following turns.
Probability: (36/84)(80/83)(79/82)(78/81)(3/80)(77/79)(76/78)(75/77)(2/76) = 0.034983%

Case 1b: If you have two identical tiles and miss one to be Three Identical Tiles. Fortunately, you pick up your target in the first turn.
Probability: (2/84) = 2.380952%

Case 1c: If you have two identical tiles and miss one to be Three Identical Tiles. But you eventually pick your target up in your third turn.
Probability: (82/84)(81/83)(80/82)(79/81)(78/80)(77/79)(76/78)(75/77)(2/76) = 2.151463%

Case 2a: If you want to have Three Tiles in Sequence or Chi but you have none of targeted tile. Fortunately, you can pick one of tiles from No. 3 to No. 7 in your first turn and then take each of their partners in following turns.
Probability: (20/84)(75/83)(74/82)(73/81)(8/80)(71/79)(70/78)(69/77)(8/76) = 0.133125%

Case 2b: If you have two tiles, which can be from No. 2 to No. 8, in sequence and miss one to be Three Tiles in Sequence. Fortunately, you pick up your target in the first turn.
Probability: (8/84) = 9.523810%

Case 2c: If you have two tiles, which can be from No. 2 to No. 8, in sequence and miss one to be Three Tiles in Sequence. But you eventually pick your target up in your third turn.
Probability: (76/84)(75/83)(74/82)(73/81)(72/80)(71/79)(70/78)(69/77)(8/76) = 4.552869%

By the above figures, the following conclusions can be drawn:

1. The probability of Chi is much higher than that of Pong however the score of Pong is not in multiple scale larger than that of Chi. It seems more efficient (by Risk to Return Ratio) to just make Chi than Pong.
2. Although the total number of tiles remained on the table will decrease along a game, the probability of taking your target will be much decreased by the possibility that your opponents pick up your target in the middle.
3. You can reduce the risk that your opponents pick up your target in your beginning turns. So if you can identify your target in the beginning, you can have higher probability to make your deal. But if you failed to do so, you will have less chance to win.
4. The probability of the target will be reduced by your opponents along the game. In contrast, if you discard a tile that is the target of an opponent, it will increase the probability that he makes his deal. Sowatch out the trend of your opponents' discarding and your own discarding.
   

Since the above estimation bases on the given assumptions, it may not give you accurate figures and the full picture of the game. If you know some about Mahjong or would like to discuss the probability question of the game, please feel free to leave your message here.