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Made hands are very rare in mahjong. Is there a table that lists the possibilities of starting with a complete hand?

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@Closers: Please let a comment. I will glad to follow your votes if I know and agree your arguments. – bigown Oct 28 '10 at 0:47
@Bigown - The question isn't clear what a "made hand" is. I'm also concerned about a flood like we had for the poker questions before... these seem like shotgun questions – LittleBobbyTables Oct 28 '10 at 0:51
@bigown I didn't vote to close, but I think this is the concern; I have posted some further discussion as well:… – Brian Campbell Oct 28 '10 at 1:45
All SE sites have to deal with vague or ambiguous questions. You ask the poster to clarify. If they do, great, if not, answer as best you can. I'm not sure 'close all vague' questions is the right response. – DaveParillo Oct 28 '10 at 15:07
I didn't know that the term "made hand" was unclear, I'm not a native speaker and know (and misused, on second thought) the term from poker. This question is asking for the chances to get a hand that allows a player to finish on their first turn. – mafu Oct 28 '10 at 15:27

Not sure I know what a 'made hand' is. Are you asking what are the odds of being dealt the Nine Gates? If yes, the odds are about 5000:1 in a 136 tile set. This page has a combinatorial treatment of the 13 orphans vs the Nine Gates.

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Just to clarify: The 5000:1 ratio described in your interesting link is not the chance to get Nine Gates but the ration between Nine Gates and 13 orphans. Both of them are, as far as I can tell, a lot rarer than 0.02%. I heard that even regular players often get them only once in their life. – mafu Oct 28 '10 at 15:26
up vote 2 down vote accepted

I wrote a program to simulate mahjong hand dealings. It's not verified yet, but so far, the ratio appears to be around 1 in 100,000 games.

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"Made" hands are those that satisfy certain requirements, specifically four "sets" of either triplets or "three in a row," plus a pair. There are also a handful of "special cases" of "made hands" that don't meet these requirements. If, after taking a tile, you don't have a "made hand" (in either regard), you need to discard a tile and try again.

The "special cases" are easily countable. The more common cases can be counted using a branch of math called "combinatorics" (the math of combinations). Increasingly powerful software programs can do this easily.

The older "authorities" probably lacked "combinatoric" capabilities, hence no tables from them. The answer (in the form of a table, or a descriptive function), could now well exist somewhere in the world. That may or may not include the western world, depending on how popular the topic of mahjong is among western problem solvers. But the answer appears attainable, and may soon be, if it hasn't been already.

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