## Yarrow Stalk Probabilities

#### by Micheal McCormick

*(Private email between the authors. Reprinted by permission.)*

In a hexagram produced by the traditional yarrow method, the probabilities
from highest to lowest are:

static yin = 424399 / 939120 =~ 45%

static yang = 4051193 / 14556360 =~ 28%

moving yang = 5957 / 26832 =~ 22%

moving yin = 77 / 1612 =~5%

(Of course these "probabilities" assume the I Ching operates randomly
through the yarrow stalks. The truth as we both know is it's a sentient
oracle.)

I seem to recall you reviewed the math behind those numbers at one
time. It was also independently reviewed by others at the request of Dick
Kaser before he used some of my work as partial basis for his book I Ching
in Ten Minutes.

Some interesting observations on the numbers (you may think of others;
wanna co-author a scholarly paper?). For instance, the probability of a
yin versus yang line in the initial hexagram (forget whether it's moving or
not) is close to 50-50:

yin = 14546989 / 29112720 =~ .4997

yang = 14565761 / 29112720 =~ .5003

So just a tiny (almost imperceptible) yang bias in the initial
hexagram. But look what happens after the moving lines change
polarity! In the second hexagram of a reading, a huge yin bias emerges:

yin = 19619714 / 29112720 =~ .6739

yang = 9493006 / 29112720 =~ .3261

Not sure what this "means".

Another application is you can pigeonhole all 64 hexagrams into 7
probability sets, Yn where 0 <= n <= 6 is the number of Yang lines in a
hexagram. Then for each set Yn, let y(n) be the frequency at which any one
of its member hexagrams would occur if the yarrow method were random:

Y0 = {hx. 64}, y(0) =~ .0155647

Y1 = {hx. 58 thru 63}, y(1) =~ .0155848

Y2 = {hx. 43 thru 47}, y(2) =~ .0156049

Y3 = {hx. 22 thru 42}, y(3) =~ .0156251

Y4 = {hx. 8 thru 22}, y(4) =~ .0156451

Y5 = {hx. 2 thru 7}, y(5) =~ .0156653

Y6 = {hx 1}, y(6) =~ 0156855

So not only are the hexagrams ordered from fewest to most yang lines, but
in the classic order they fall neatly into 7 sequential sets whose
frequencies graph an almost perfect bell curve. (The 20 hexagrams of Y3
are the largest set and have the highest probability.)