### The Probabilities of the Lines: Yarrow

#### Probabilities of the four hyperlines

Each hyperline is determined by three divisions of the heap into two piles.

#### First division (of three)

The original heap has 50 stalks but one is set aside immediately and then plays no further role. Thus, in the first division, the heap has 49 stalks. After the division, one stalk is removed from the right pile. Then the left pile is reduced mod 4, then the right pile is reduced mod 4. Setting the discarded stalks aside, and counting the one stalk removed after the division, there remain:
```	1 + 4 + 4 = 9 stalks (code = 2)
1 + 3 + 1 or
1 + 2 + 2 or
1 + 1 + 3 = 5 stalks (code 3)
```
So the probability of code 2 is 1/4, and of code 3, 3/4.

#### Second division (of three)

In the second division the heap consists of the discards after the first division: 49-9 = 40, or 49-5=44 stalks. In either case, the heap is divisible by 4, so after the division, removing one stalk from the right pile, and reducing each pile mod 4, there remain:
```	1 + 3 + 4 or 1 + 4 + 3 = 8 (code = 2)
1 + 1 + 2 or 1 + 2 + 1 = 4 (code = 3)
```
which each occur with probability 1/2.

#### Third division (of three)

The third division is similar to the second. Adding the codes for each of the three divisions, we obtain the hyperline codes:
```	2 + 2 + 2 = 6 (probability 1/4 * 1/2 * 1/2 = 1/16)
(the probability of hyperline 6 is 1/16)

2 + 2 + 3 = 7 (prob. 1/4 * 1/2 * 1/2 = 1/16) or
2 + 3 + 2 = 7 (prob. 1/4 * 1/2 * 1/2 = 1/16) or
3 + 2 + 3 = 7 (prob. 3/4 * 1/2 * 1/2 = 3/16)
(the probability of hyperline 7 is 5/16)

2 + 3 + 3 = 8 (prob. 1/4 * 1/2 * 1/2 = 1/16) or
3 + 2 + 3 = 8 (prob. 3/4 * 1/2 * 1/2 = 3/16) or
3 + 3 + 2 = 8 (prob. 3/4 * 1/2 * 1/2 = 3/16)
(the probability of hyperline 8 is 7/16)

3 + 3 + 3 = 9 (prob. 3/4 * 1/2 * 1/2 = 3/16)
(the probability of hyperline 9 is 3/16)
```
All this is explained in detail in Wilhelm, pp. 721-722. In summary:
 6 -x- old yin 1/16 7 --- young yang 5/16 8 - - young yin 7/16 9 -o- old yang 3/16

This is radically different from the coin oracle, and this is just one reason for preferring the yarrow method.

In practice, an experienced hand will not achieve these probabilities, for the reason described above. In avoiding the very unequal divisions, a small advantage is gained to the remainder 8 and its score 2, and so the expectation of a 6 line, old yin, will be a bit larger than 1/16. This effect is included in our simulation by the use of a chaotic attractor to arrange the heap. By a series of experiments, you may choose a heap algorithm to match your own hand.

#### Line probabilities in the first hexagram

Casting a hyperhexagram with 18 divisions of yarrow determines two hexagrams. A yang line in the first hexagram results from either a young yang or an old yang hyperline in the hyperhexagram. Thus the probability of an initial yang line is the sum of the probabilities of old yang (3/16) and young yang (5/16) or 8/16: 50%.

The chances of an initial yin line are similarly the sum of old yin (1/16) plus young yin (7/16), or 8/16: 50%. Initially, yang and yin are equiprobable. This is the same as the coin oracle, in which initial yin and yang are also balanced 50-50.

#### Probabilities for changing lines

The chances of a changing yarrow hyperline are (1/16) for old yin, plus (3/16) for old yang, or 4/16: 25%. Again, this is the same as in the coin oracle.

#### Line probabilities in the second hexagram

In the second hexagram, a yin line results from either an original young yin (7/16) or an old yang (3/16) or 10/16 = 5/8. Similarly, a yang line results from either an initial young yang (5/16) or an old yin (1/16) pr 6/16 = 3/8. Here we have a significant difference between the yarrow-stalk oracle and the coin oracle.

#### Summary

In first hexagram, the yin and yang lines are equally probable, with both the yarrow and the coin oracles.

After the changes, yin and yang lines are equally probable with the coin oracle. But with the yarrowstalk oracle, yin lines are 5/3 times more likely than yang. Yang changes to yin more frequently than yin to yang.

This is the reason for thinking that the coin oracle has contributed to world problems. Yarrow seems to be a relic of prepatriarchal times.

Revised 17 July 2003 by Ralph