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I Ching - Probabilty: Yarrowstalks vs coins

Jan 23, 1994 01:48 PM
by John Mead

                         Comments on the I Ching
                                by John Mead

   Although the I Ching is usually regarded as a manual of divination,
the Neo-Confucians used it as a work of profound metaphysical
significance.  In particular, the school of "Form and Numbers" was
centered around the philosopher Shao Yung (1101-1177). Yung conceived
that the universe was constructed from numerical progressions of 4.
The number 4 corresponds to the cosmic forces of the Greater Yang,
Greater Yin, Lesser Yang, and Lesser Yin.  From these, all nature is
constructed (the seasons, the elements, etc.).

   The traditional Yarrow stalk approach to divination is based on
this number 4.  Fifty stalks are first taken in hand, less one cast
aside (unused).  The 49 stalks are then randomly divided into two piles.
One then takes a single stalk from the right hand pile, and places
it between the ring and last finger of the left hand.  The left hand
pile is placed in the left hand, and groups of 4 stalks are removed
until there are 4 or fewer stalks remaining.

   These stalks (1,2,3,or 4) are placed in the right hand between the
ring and last finger.  The right hand pile is now taken in the left
hand, where once again groups of 4 are removed until 4 or fewer
remain.  Surprisingly, there are only four combinations of remaining
stalks: 1+4+4, 1+3+1, 1+2+2, 1+1+3. (The sum of the two remainders
must be divisible by four).

   Each of these combinations are equally likely.  The first group (9)
is designated as Yin and given the number 2. The other groups
(totalling five) are designated as Yang and assigned the number 3.
These stalks are now set aside, and the remaining stalks are
randomly divided into two groups (note that this remaining group
is formed from the clumps of 4's which we removed; hence the total
number of stalks we are dealing with is a multiple of four).  Again we
take one stalk from the right hand pile and place it between the
ring and last finger of the left hand. Grasping the left hand
pile, we again remove multiples of 4 stalks until 4 or fewer
remain. The right hand pile is also counted as before. The four possible
(equally likely) combinations are now: 1+4+3, 1+3+4, 1+2+1, 1+1+2 (the
two remainders must total one less than 4 or 8, i.e., 3 and 7). The
first two possibilities (totalling 8) are Yin and assigned the value
of 2. The other two options (totalling 5) are Yang and are assigned
the value of 3. Again, we set these stalks aside and take the
remaining ones, randomly divide them into two piles, and count
as before (the leftover stalks again total to a multiple of 4).
We again end up with combinations (equally likely) as
before: 1+3+3, 1+3+4, 1+2+1, 1+1+2.  Values are assigned as before.

The above procedure is repeated to form each of the six lines of
the I Ching Hexagram.  The three results are totaled to form a single I
Ching line as follows:


1st   + 2nd    + 3rd   = Total
2(9)  + 2(8)   + 2(8)  = 6  (old) Greater Yin
3(5)  + 2(8)   + 2(8)  = 7  (young) Lesser Yang
3(5)  + 2(8)   + 2(8)  = 7  L. Yang
3(5)  + 2(8)   + 2(8)  = 7  L. Yang

2(9)  + 3(4)   + 2(8)  = 7  L. Yang
3(5)  + 3(4)   + 2(8)  = 8  (young) Lesser Yin
3(5)  + 3(4)   + 2(8)  = 8  L. Yin
3(5)  + 3(4)   + 2(8)  = 8  L. Yin

2(9)  + 2(8)   + 3(4)  = 7  L. Yang
3(5)  + 2(8)   + 3(4)  = 8  L. Yin
3(5)  + 2(8)   + 3(4)  = 8  L. Yin
3(5)  + 2(8)   + 3(4)  = 8  L. Yin

2(9)  + 3(4)   + 3(4)  = 8  L. Yin
3(5)  + 3(4)   + 3(4)  = 9  (old) Greater Yang
3(5)  + 3(4)   + 3(4)  = 9  (old) Greater Yang
3(5)  + 3(4)   + 3(4)  = 9  (old) Greater Yang

The Young Yang is designated    -------
The Young Yin is designated     --- ---

The Greater Yang and Greater Yin are designated similarly.
However, they reverse (Yin -> Yang and Yang -> Yin) to form
the reading of the "Change" or second hexagram.

   The probabilities of the Yarrow stick method are:

    1 out of 16  G. Yin
    3 out of 16  G. Yang
    7 out of 16  L. Yin
    5 out of 16  L. Yang

   Another method of generating the  Hexagrams is by using the coin toss
method.  The 3 coins are assigned values as follows:

     Heads = 2, Tails = 3

   The 3 coins are totaled and valued as above per each hexagram line.
The probabilities for this method are:

G.   Yang (3+3+3=9) (1/2)*(1/2)*(1/2) = 1 out of 8
G.   Yin  (2+2+2=6) (same as above)   = 1 out of 8
L.   Yin  (2+3+3=8) (tth,tht,htt)     = 3 out of 8
L.   Yang (2+2+3=7) (hht,hth,thh)     = 3 out of 8

   These two methods of creating the I Ching hexagrams are not equal!!
To match the odds of the Yarrow stick method, one can do the following.
Toss two coins, and separate out the one with tails on it. If both
are heads, leave one as heads. Then toss the remaining 2 coins
and add the *3* coins as usual, to form a single line of the hexagram.
With this method, the first toss has results of  (tt,ht,th) 3/4 = 3
Yang and (hh) 1/4 = 2 Yin.  The second toss has results of
(hh,ht,th,tt) 1/4 for a 6 value, 1/4 for a 4 value and  1/2 for a 5
value. Hence we would get a total probablility for each line of:

G.   Yin  (2+4)  (1/4) * (1/4)  or 1/16
L.   Yin (3+5)or(2+6):(3/4)*(1/2)+(1/4)*(1/4)
          or (3/8)+(1/16) = 7/16.
G.   Yang (3+6):(3/4)*(1/4)  or 3/16
L.   Yang (3+4)or(2+5):(3/4)*(1/4)+(1/4)*(1/2)
          or (3/16)+(1/8) = 5/16

   This matches the Yarrow stalk method.

   As Jung said in the introduction of Richard Wilhelm's translation
of the I Ching, "The validity of synchronicity [probability] is the
observer's opinion that the text of the hexagram amounts to a true
rendering of his psychic condition".  Hence, one might ask if it
matters at all which method we use.

   I propose each individual should ask this question of the I Ching
itself, and arrive at one's own opinion.  Who better to ask??

Peace --

John Mead

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