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Mahjong and Probability

When people talk about building a hand in mahjong, some may think the main question is "what yaku should I make?" I think that misses the essence of mahjong.

The essence of hand-building is to make the shape of four sets and one pair.
If you look at mahjong one hand at a time,
it is fundamentally a game of which of the four players completes four sets and one pair first.

Of course the final evaluation is still based on points, but modern mahjong has many dora, including red fives, and high scores are common even without elaborate yaku. Given that game structure, the essence is still not "yaku" but "shape."

And since only the player who completes the hand shape first gets the points, speed naturally becomes essential.

So how do you win faster? The way of thinking needed for that is tile efficiency and probability.

Example 1: Which Tile Should You Discard?

七万牌图八万牌图一索牌图二索牌图三索牌图五索牌图五索牌图一筒牌图三筒牌图四筒牌图　　Pon中牌图中牌图中牌图　　　Tsumo九万牌图

Let us start with the simplest possible example.
In Example 1, three complete sets and the pair are already finished, so you only need one more set.

If you cut 一筒牌图, then drawing or being given 二筒牌图 or 五筒牌图 wins the hand.

If you cut 四筒牌图, then only 二筒牌图 completes the hand.

It is obvious which choice gives the better chance to win.

Example 2: Which Tile Should You Discard?

七万牌图八万牌图一索牌图二索牌图三索牌图五索牌图五索牌图三筒牌图四筒牌图四筒牌图　　Pon中牌图中牌图中牌图　　　　　Tsumo九万牌图

The two discards that let you take tenpai, a state where you are one tile from winning, are 三筒牌图 and 四筒牌图.

If you cut 三筒牌图, your wait becomes 五索牌图 and 四筒牌图.

If you cut 四筒牌图, your wait becomes 二筒牌图 to 五筒牌图.

At first glance both waits seem to involve two kinds of tiles, but that does not mean the chance of winning is the same.

You are already using two copies each of 五索牌图 and 四筒牌图, so only four tiles remain in total.

By contrast, if you judge only from your own hand, all eight copies of 二筒牌图 and 五筒牌图 are still alive.

If you were drawing lottery tickets, you would obviously prefer the side with more winning tickets.

So if you take the wait with more tiles, 二筒牌图 to 五筒牌图, the probability of winning is naturally higher.

Comparison diagram of acceptance counts for two tenpai choices

"Choose your discard so that you enter the draw with the most favorable odds."
That is the most basic way to think about hand-building.

七万牌图八万牌图一索牌图二索牌图三索牌图五索牌图五索牌图三筒牌图四筒牌图四筒牌图　　Pon中牌图中牌图中牌图　　　Tsumo九万牌图

For example, even if you cut 四筒牌图 here, you may still see 五索牌图 or 四筒牌图 come out first and miss the win.

Even so, do not think, "I should have cut 三筒牌图 instead." Mahjong is full of cases where a bad play still happens to produce a good short-term result.

There is no reason to obsess over that kind of purely results-based regret.

Summary and Principle

Mahjong is a game in which even the best play will often fail. Do not swing emotionally with each short-term result. Instead, judge which choice is better over the long run.

Original Japanese page: http://beginners.biz/kihon/kihon02.html

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