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December 31, 2004
Cross the Great Stream
It's obvious to anyone who's tried it that the yarrow-stalk method of consulting I Ching is superior in terms of the production of a meditative state through the routine automatism of the fairly long and convoluted process of drawing stalks. The result is 'seeded' right at the beginning, in the sense that you could count the number of stalks in the two piles and immediately tell the result. However the long drawing-out of the consequences of the random/unconscious division of the stalks seems as important; there's an interesting sense in which the automatism of following the trail seeded by the division, in turn 'seeds' the contemplative state necessary to make the next division.
Anyway, I'd wondered for a long time about the probabilities involved: it seems obvious that the stalks are a more complex procedure than coins and that the chances of getting the different lines are not identical. Luckily, before being tempted into what would inevitably have been another lengthy bout of painful number-crunching, I found this summary which shows the probabilities as follows:
-x- Yarrow : 1/16 Coins 2/16
--- Yarrow : 5/16 Coins 6/16
- - Yarrow : 7/16 Coins 6/16
-o- Yarrow : 3/16 Coins 2/16
Obviously, the stalks are far more 'weighted', and arguably more interesting for it; the slight disparity between the 7 and 8 (--- and - -) lines, and the radical disparity in the chances of getting the two moving lines.
It's easy to work out that (not including moving lines, that's too much to do by hand :) there are four composite probabilities of getting the trigrams : 125/4096 (all 7s), 343/4096 (all 8's) 245/4096 (two 7s one 8) and 175/4096 (two 8s one 7)
___
___ Chhien = 5/16 * 5/16 * 5/16 = 125/4096 001111101 / 100000000000
___
- -
- - Khun = 7/16 * 7/16 * 7/16 = 343/4096 101010111 / 100000000000
- -
- -
- - Chen = 7/16 * 7/16 * 5/16 = 245/4096 011100001 / 100000000000
---
- -
--- Khan = 7/16 * 5/16 * 7/16 = 245/4096 011100001 / 100000000000
- -
---
- - Kên = 5/16 * 7/16 * 7/16 = 245/4096 011100001 / 100000000000
- -
---
--- Sun = 5/16 * 5/16 * 7/16 = 175/4096 010101111 / 100000000000
- -
---
- - Li = 5/16 * 7/16 * 5/16 = 175/4096 010101111 / 100000000000
---
- -
--- Tui = 7/16 * 5/16 * 5/16 = 175/4096 010101111 / 100000000000
---
The composite probabilities of the hexagrams, then:
Chhien = 125/4096 * 125/4096 = 15625 / 16777216
Khun = 343/4096 * 343/4096 = 117649 / 16777216
Chun = 175/4096 * 175/4096 = 30625 / 16777216
Mêng = 175/4096 * 175/4096 = 30625 / 16777216
Hsü = 175/4096 * 245/4096 = 42875 / 16777216
Sung = 125/4096 * 175/4096 = 21875 / 16777216
Shih = 343/4096 * 175/4096 = 60025 / 16777216
Pi = 343/4096 * 175/4096 = 60025 / 16777216
Hsiao Hsü= 245/4096 * 125/4096 = 30625 / 16777216
Li = 125/4096 * 245/4096 = 30625 / 16777216
Thai = 343/4096 * 125/4096 = 42875 / 16777216
Phi = 125/4096 * 343/4096 = 42875 / 16777216
Thung Jen= 125/4096 * 245/4096 = 30625 / 16777216
Ta Yu = 245/4096 * 125/4096 = 30625 / 16777216
Chhien = 343/4096 * 175/4096 = 60025 / 16777216
Yü = 175/4096 * 343/4096 = 60025 / 16777216
[....TBC.....next year, LOL!]
By pairing up the trigrams, you'd expect there to be more distinct probabilities, but surprisingly there are fewer because 175*175=245*125. But you can see that pairing up the trigrams results in huge disparities, of an order of magnitude, in the likelihood of getting (to take the worst instance) Chhien and Khun. More generally the slightly smaller probability of getting a broken line means that the more a hexagram is dominated by these receptive/female lines the less likely it is to turn up (this takes no account of the different meanings of the relative positions of the lines, of course)
And am I wrong in thinking that these weird-looking probabilities are undecomposable into simpler fractions?
[UPDATE : No, it's true - at least for those hexagrams above, there are no common factors]
None of this helps me work out what the hell the oracle is talking about.
Posted by undercurrent at December 31, 2004 10:51 PM