Mah-Jong is a Game of Primarily Sequences

by Alan Kwan


Every mah-jong player knows that it is, in general,
easier to make a sequence than a triplet.  A question
few have asked is, easier /by how much/?  This article
on simple Combinatorics attempts to offer some insights 
on the question.  Some background in elementary Combinatorics
of the reader is preferred.  (Please correct any mathematical 
errors that may be here.)

One way of looking at the relative ease of making sets
is to count how many such sets are contained in the 136-tile
(counting only playable tiles) deck.  (Note that this works
/only/ when comparing sets which consist of the same number
of tiles, namely 3 in this case.)  This is what will be 
done in this article.

Let's begin with sequences.  There are 3 suits, and there
are 7 possible sequences, from 123 to 789, in each suit.
To construct each sequence, you take 1 out of the 4 tiles
for each of the 3 numbers.  Therefore, there are

s = 3 * 7 * (4^3)

possible 3-tile combinations in the 136-tile deck that are
sequences.

Now, let's look at triplets.  There are 34 playable tile
types.  To make a triplet, you take 3 tiles out of the 4
of a tile type.  There are C(4,3) = 4 ways to do the
selection.  Therefore, there are

t = 34 * 4

possible 3-tile combinations in the 136-tile deck that are
triplets.

Dividing s by t, we get

s / t = 9.88...

Which means that there are about 10 times as many sequences
as triplets in the deck!

Of course, this figure is only part of the story.  The exact
mechanisms of play affects how sequences and triplets are
formed.  Also, there is the important distinction that you
can "chow" from only your upper seat, while you can "pong"
from anybody.  

The reader should be careful not to misinterpret this figure.
All it means is that, if you take 3 tiles (or, as a 
reasonable estimate, a few tiles) randomly from the
deck, it is about 10 times more likely that you get a sequence
than a triplet.  This figure does not represent your chances
of /completing/ a set based on partial elements in the hand.

Another item of interest is the last terms in the above formulae
for s and t.  There are 4^3 = 64 ways to make any specific
sequence, but only 4 ways to make any specific triplet, out
of the 136-tile deck.  The difference is a factor of 16,
not (16/6) as someone will think when considering only
the effects of "fewer tiles of the type left when getting the
second and third tiles of a triplet".  (Again, the reader should
be careful not to mis-apply this figure to completing partial
sets.)

In conclusion, this is what we can safely and soundly claim: 
there are about 10 times as many sequences as triplets in the 
deck.  How this is to be interpretated to answer the opening
question, however, is left to the readers and players.