The Probabilities of the 4096 Changes:
Yarrow Oracle

We have seen that the yarrow probabilities of the four hyperlines are:
6 -x- old yin 1/16
7 --- young yang 5/16
8 - - young yin 7/16
9 -o- old yang 3/16

After casting a hyperhexagram (a change) with three divisions of a heap of yarrow stalks, we may record the result as a hexcode. For example, "678969" means a six (old or changing yin) in the bottom line, a seven (young or unchanging yang) in the second line, and so on, with a nine (old or changing yang) on the top.

If this change is obtained by choosing two hexagrams at random, then its probability is 1/4096, or 0.000244140625, as all 4096 changes are equally likely.

But if the same change is obtained by three divisions of yarrow stalks, the probability is different. We may calculate the probability of this change from its hexcode, "678969", by multiplying the probability of the hyperline codes in order,

p(6) * p(7) * p(8) * p(9) * p(6) * p(9) =
1/16 * 5/16 * 7/16 * 3/16 * 1/16 * 3/16 =
1 * 5 * 7 * 3 * 1 * 3 / 16 ^ 6 =
315 / 2 ^ 24 = 315 / (4096 * 4096) =
315 / 16,777,216
where p(6) denotes the probability of obtaining a six, and so on. In other words, using yarrow stalks this change is (315/4096) or about 0.08 times as likely as opening the book twice at random. We will refer to this factor, (315/4096), as the "yarrow factor" of the change. Thus the yarrow probability of a change is the yarrow factor times the random probability.

From this example we may derive a general rule for the yarrow factor of a change: let n6 be the number of sixes in the hexcode of the change, n7, n8, and n9 similarly, the number of sevens, eights, and nines in the hexcode. Then the yarrow factor is given by the formula:

5^n7 * 7^n8 * 3^n9 / 4096

Note: The yarrow factor ranges from a minimum (in the case of all six lines being sixes) of 1/4096 (corresponding to a very small probability of about 6 times in a hundred million) to a maximum yarrow factor (in the case of all six lines being eights) of

7^6 / 4096 = 117,649 / 4096 = 28.722900390625
corresponding to a probability of about 7 times in a thousand.

WE may think of the yarrow factor of a casting as varying from about zero (extremely rare) through 29 (relatively common).

Revised 18 Dec 2007 by Ralph