I Ching and probabilities
One can, with a certain provocation, to summarize the pulling of I Ching to a procedure allowing to obtain a random series. The technique of pulling with the coins in is revealing. Pulling with the rods of yarrow, although more complex, also underlies a game of chance to the well-known laws.
To apprehend Yi-King under the mathematical angle misses a little poetry, but does not betray of anything its fundamental principles. On the contrary, this angle opens on a perception fractale world. The whole is in one, even if we cannot circumscribe it (this is why a first rod is isolated in precondition to the first transformation of the first line). A way of describing this recursivity, this overlap - with the manner of the Russian headstocks - is to consider the world complexes the human one, by means of figures drawn from another sphere; by respecting certain proportions. Just like the golden section revealed the beauty in the fabrics of Masters, the rods deliver 6 magic numbers to us forming a key.
Basic probability: The yin and the yang
Pulling by the coins gives the following probabilities:
Coins Probability Line Comment
3 faces 1/8 Old yang ready to become a Young Yin
2 piles 3/8 Young Yang
2 faces 3/8 Young Yin
3 piles 1/8 Old yin ready to become a Young Yang
We can deduce from it that the probability of the yin is equal to that of the yang: 3/8+1/8=1/2
The probability of obtaining a mutant line is of 2/8 = 1/4
Pulling by yarrow is more subtle:
Yarrow ProbabilitÃ© Ligne Commentaire
9 heaps of 4 3/16 Old yang ready to become a young Yin
7 heaps of 4 5/16 young Yang
8 heaps of 4 7/16 young Yin
6 heaps of 4 1/16 Old Yin ready to become a young Yang
One can deduce from it that the probability of the yin is equal to that of the yang
5/16+3/16 = 7/16+1/16 = 1/2
On the other hand, the probability of obtaining a mutant line yang is different from the probability of obtaining a mutant line yin (3/16 against 1/16). What clearly differentiates pulling by yarrow from pulling by the rods ! On the other hand, the general probability to obtain a mutant line is the same one (1/4)
Probability of a change
As we saw, the change has a chance on four to appear.
From there, which is the probability of obtaining N changes? (N between 0 -no lines change- and 6 - all the lines are changing-)
The answer is to be found in the binomial distribution.
Which is the probability of exit of an unspecified hexagram with 2 changes?
One can combine two mutants out of 6 in several way (line 1 & 3, or line 2 & 6, etc). The order does not have importance. It is thus about a combination (any encyclopaedia gives the formula of it):
CnN = Factorial N/(Factorial (N-n) * Factorielle(n))
Where N is the number of tests (for Yi King, there are 6 lines!) and N the number of studied success (a mutant number of line of the hexagram).
That only indicates to us, how much combination can be carried out. But not yet probability!
It is necessary for that to introduce:
CnN * [ p exposing (N)) * [ (1-p) exposing (N-n) ]
Where p is the probability of exit of required pulling (here, a mutant line, is 0.25)
We have :
Probability of exit of a hexagram with two mutant lines = 15 * 0.0625 * 0.31640625)=0.29663086 which is exactly the result of Binomiale(N;p)
Probability of a hexagram
All hexagram thus has the same probability of emerging from the interrogation by Oracle.
Since Yin & Yang appear with each one probability 1/2 and that pullings are independent. It is thus 1/64!
But as we showed, the changes come to disturb this too simple mechanics, too symmetrical. They introduce a small disordered state of which I like to hear resonances in the field of physics. Analogical bullshit ? I return you being studied on the question of symmetry in the universe. To study to be convinced "the end of the parity" in physics in 1957.
Moreover, a small trick: believe that the symbol of the yin and yang is symmetrical?
It is not. One cannot superimpose his image with itself in a mirror.