Yarrow stalks

Yarrow stalks have been used in China for divination since ancient times. There are reference of yarrow stalks divinations being used  to get a "second opinion" on important matters for which divination with cracks in tortoise shells were already be perfomed.

Unfortunately, the method that was used in such ancient times has been lost. The procedure we use today is a reconstruction dating back the 12th century CE and is described in the commentaries that form the Ten Wings.

This method is quite laborious and requires some dexterity (to hold the yarrow stalks) and focus (to properly count them). Its complexity is both its strong and weak point: some find it too bothersome while others consider the time required to be well spent as they can meditate on the question.

The process starts with 50 yarrow stalks of which one is put aside and no longer used.
To get a line one has to proceed as follows:

1. Split the 49 stalks in two groups;
2. Take one stalk from the left group and put it aside;
3. Count the left group by four until you have four or less stalks left in the group;
4. Put the (one to four) remaining stalk together with the one you took on step 2;
5. Count the right group by four until you have four or less stalks left in the group;
6. Put the (one to four) remaining stalk together with the ones you got from step 2 and 5;
7. If you remained with nine stalks, mark 2, if you remained with five stalks mark 3;
8. Put the stalks you counted all together (they should be 40 or 44), split them in two groups and repeat steps 2-6;
9. If you remained with eight stalks, mark 2, if you remained with four stalks mark 3;
10. Put the stalks you counted all together, (they should be 32, 36 or 40) split them in two groups and repeat steps 2-6;
11. If you remained with eight stalks, mark 2, if you remained with four stalks mark 3;
12.  Sum up the three numbers you got, the sum should be either 6, 7, 8 or 9, and draw the line according the following table.
    

    6	7	8	9

Repeat the entire procuder other five times drawing the lines from the bottom to the top to get the hexagram.

An alternative method of counting is to ignore steps 7, 9 and 11 and group all the stalks you get in a single heap. After you have performed the split three times, you divide the stalks in the heap (which will contain eather 24, 28, 32 or 36 stalks) by four to directly get the number of the resulting line: 6, 7, 8, 9.

Actually this can be simplified further to avoid counting both groups. After step 3:

• if you get four stalks, pick four stalks from the other group and count 2 
• if you get three stalks, pick one stalks from the other group and count 3 
• if you get two stalks, pick two stalks from the other group and count 3 
• if you get one stalks, pick three stalks from the other group and count 3  

since you have to end up with either nine or five stalks (counting the first one you set aside).

After step 8 (and 10):

• if you get four stalks pick three stalks from the other group and count 2 
• if you get three stalks pick four stalks from the other group and count 2 
• if you get one stalks pick two stalks from the other group and count 3 
• if you get two stalks pick one stalks from the other group and count 3

since you have to end up with either eight or four stalks (counting the first one you set aside).

Searching on YouTube will provide you with a great deal of video example on how to use the Yarrow stalks to get hexagram lines.

Probabilities

The probabilities for this method are usually considered to be:

Prob(6) = ¹/₁₆
Prob(8) = ⁷/₁₆
Prob(7) = ⁵/₁₆
Prob(9) = ³/₁₆
Prob(yin) = Prob(yang) = ¹/₂

on the basis of the following reasoning:

• On the first subdivision, 49 stalks, we can get 2 with a probability of  ¹/₄ and 3 with a probability of  ³/₄
• On the second and third subdivision, we can get 2 with a probability of  ²/₄ and 3 with a probability of   ²/₄
• Hence the probabilties for each possible outcome are:

  Prob(2+2+2)	= 1/4 * 1/2 * 1/2 = 1/16
  Prob(2+2+3)	= 1/4 * 1/2 * 1/2 = 1/16
  Prob(2+3+2)	= 1/4 * 1/2 * 1/2 = 1/16
  Prob(3+2+2)	= 3/4 * 1/2 * 1/2 = 3/16
  Prob(2+3+3)	= 1/4 * 1/2 * 1/2 = 1/16
  Prob(3+2+3)	= 3/4 * 1/2 * 1/2 = 3/16
  Prob(3+3+2)	= 3/4 * 1/2 * 1/2 = 3/16
  Prob(3+3+3)	= 3/4 * 1/2 * 1/2 = 3/16

• Summing up the probabilities for each possible result, we get:

  Prob(6) =	Prob(2+2+2)		= 1/16
  Prob(8) =	Prob(2+3+3) + Prob(3+3+2) + Prob(3+2+3)	= 1/16 + 3/16 + 3/16	= 7/16
  Prob(7) =	Prob(2+2+3) + Prob(2+3+2) + Prob(3+2+2)	= 1/16 + 1/16 + 3/16	= 5/16
  Prob(9) =	Prob(3+3+3)		= 3/16

Unfortunately, the analysis above is not accurate as it assumes that, for each subdivision, the four possible outcames are all equiprobable, which is not the case.

Expecially for the first step:If we assumed all the possible split of 49 stalks to be equiprobable, the chance to get a 2 would be ¹¹/₄₇, meaning that getting a 6 as final outcome would have a probability of  1,28% which is much lower than ¹/₁₆ (6.25%).

However, how "random" would you consider a split where one group would contain just one stalk? Not much, I guess.

In fact, the closer the 49 stalks are split in the middle, the closer the chance of getting 2 approximates ¹/₄ (and hence the probability of getting 6 get closer to  ¹/₁₆).

This is true for the second and third subdivision as well, but the effect is not very relevant and the probability to get 2 or 3 are really ¹/_{2 .} 

This leads to the interesting conclusion that we cannot tell the exact probability distribution of the yarrow stalks method:

With the yarrow stalks method, the probability of getting 6, 7, 8, and 9 is, respectively, ¹/₁₆, ⁵/₁₆, ⁷/₁₆ and ³/₁₆ as much as the heaps of stalks are split very close to the middle of the heap.
The actual probabilities depend on the way each split is done and varies, even if only slightly, at each cast.