Warring States Project Arithmetic

It is not difficult (though it is perhaps a little tedious) to see why the Chu Dau/Dv Jing text result is decisively in favor of the Project's accretional theory of that text. We will work through the arithmetic on this page. Those prepared to take the demonstration for granted may return to the Summary page by clicking HERE.

The Problem

In previous lectures (from 1993 and following) and publications (going back to 1994), we had put forth an accretional theory of the Dau/Dv Jing, according to which the work we gave grew by continuous additions from the middle of the 04th century to the middle of the 03rd century. If that process could be observed at the point it had reached by sometime in the early 03rd century, we would have a check on this hypothesis. The Gwodyen 1 tomb, both archaeologically and contextually, is from the early 03rd century (it cannot be later than 0278, the year Chu abandoned its nearby capital, and moved east). The hypothesis thus predicts that the DDJ would have been still incomplete when it was drawn on by a Chu scribe to produce the Gwodyen tomb text. What is the actual condition of things with the Gwodyen text? It is a selection of 33 passages, represents 32 different chapters of the DDJ, but none of those chapters are from the fifteen-chapter span at the end of our present work, DDJ 67-81. The selection thus seems to have been drawn from a not yet complete DDJ, one which had reached DDJ 66 (or perhaps a little further), but not yet DDJ 81, the present end of the work. This on its face strikingly confirms the hypothesis.

In response, it had been argued by some that the DDJ was complete, in 81 chapters, when the Chu selection was made from it in c0288, and that the omission of the last 15 chapters is due to chance. Is this likely? More to the point, because it can be objectively tested: Is it statistically possible?

The standard statistics textbook would put it this way. An urn contains 66 White balls and 15 Red balls (total: 81 balls). Without looking, we draw 33 balls from the urn, and in all but one case we do not replace the ball once it has been drawn. Under those conditions, what is the chance that we will draw only White balls, and no Red ball?

The Arithmetic

The task of drawing only White balls in 33 draws, at first looks promising. On the first draw, we have 66 chances out of 81 to get a White ball. That is a decimal probability of 66/81 = 0.8148. That ball is not replaced, and so as we make our second draw, there are now 65 White balls out of a total of 80. The chance of getting White is thus a little less: 65/80 = 0.8125. But it is still better than half; in fact, drawing a White ball on this step is obviously the more likely outcome. And so it goes at each step: The total number of balls is reduced by 1, and the number of White balls is also reduced by 1, and so the fraction representing the chance of drawing a White ball at any given step goes slowly but steadily down.

With sequential events, probabilities multiply. We will thus be multiplying a series of 33 fractions together, to get the net probability of 33 white balls in a row. For clarity, we show in the table below the results at each step of the process, giving both the current chance of getting a White ball on that draw, and also (as "Cum") the cumulative chance of getting a White ball on this and all preceding draws together. It is the cumulative figure which counts.

Here we go:

Draw	1	2	3	4	5	6	7	8	9	10
White	66	65	64	63	62	61	60	59	58	57
Total	81	80	79	78	77	76	75	74	73	72
Chance	0·8148	0·8125	0·8101	0·8077	0·8052	0·8026	0·8000	0·7973	0·7945	0·7917
Cum	0·8148	0·6620	0·5363	0·4332	0·3488	0·2800	0·2240	0·1786	0·1419	0·1123

At the beginning, as noted, the odds favor drawing White; on the first draw, the exact chance is 0.8148, or about 4 out of 5. But already at the second turn, the cumulative chance of getting two White balls in a row is down to 0.6620, or roughly 2 out of 3. By the fourth turn, the chance of having drawn only White balls to that point is down to less than half. That is, it would be just a hair more likely to have drawn 1 Black ball somewhere among the first 4 draws. Still, there is nothing yet that will raise serious suspicions. If we persist to the 10th turn, at the end of the row, the probability of having drawn only White balls is down to 0.1123, or a little better than 1 in 10. That gives roughly a 90% degree of certainty that something nonrandom is happening. But the 90% level is not generally recognized as a decisive level of certainty. We may thus say that if the Gwodyen florilegia had contained only 10 DDJ extracts, all of them from the White or DDJ 1-66 part of the source text, though our hypothesis of an incomplete DDJ text is strongly supported, it cannot be said to be decisively confirmed.

But the Chu text is larger than this, and thus the count goes on. Here are the chances that the next 10 draws will also produce only White balls, and no Red balls:

Draw	11	12	13	14	15	16	17	18	19	20
White	56	55	54	53	52	51	50	49	48	47
Total	71	70	69	68	67	66	65	64	63	62
Chance	0·7887	0·7857	0·7826	0·7794	0·7761	0·7727	0·7692	0·7656	0·7619	0·7581
Cum	0·0886	0·0696	0·0545	0·0425	0·0330	0·0255	0·0196	0·0150	0·0114	0·0087

By the end of this series, we have reached a cumulative probability of slightly less than 1 in 100; that is, we are past the the level of 99% confidence that the result in question could not have been produced by chance. By our working rule (at a level of 99%), that amounts to operative certainty. The calculations up to this point are enough to suggest that the Chu text preponderance cannot reasonably be attributed to chance.

But the text itself continues, and so shall we. Here is the third set of ten draws:

Draw	21	22	23	24	25	26	27	28	29	30
White	46	45	44	43	42	41	40	39	38	37
Total	61	60	59	58	57	56	55	54	53	52
Chance	0·7541	0·7500	0·7458	0·7414	0·7368	0·7321	0·7273	0·7222	0·7170	0·7115
Cum	0·0065	0·0049	0·0037	0·0027	0·0020	0·0015	0·0011	0·0008	0·0006	0·0004

The exact cumulative probability as of the 30th consecutive White draw, to the limits of our small calculator, is 0.0003915, or about 1 in 3,000. This is far beyond the tables of probability as given in the standard textbooks. That is, in terms of normal statistical interpretation, we are more than certain of the correctness of our hypothesis.

Still, the text itself continues a little while longer, so here is the final set of three draws. The last fraction, which is the same as on the previous draw, represents a draw with replacement. This is required for accuracy, since 1 of the 33 Gwodyen chapters duplicates another. The difference in the total calculation is extremely small, but precision is better in an argument on which important conclusions depend.

Draw	31	32	33
White	36	35	35
Total	51	50	50
Chance	0·7059	0·7000	0·7000
Cum	0·0003	0·0002	0·0001

Up to the very end, as we see from the "Chance" row of the table, the chance of drawing a White ball on that one draw is in the vicinity of 7 out of 10, which is still a quite likely proposition. It never happens, in the whole sequence, that the chance of drawing Black on any one trial is higher than that of drawing White. But the claim we are examining is that we have a reasonable chance of getting a White ball on that and every preceding draw. The chance of that outcome has declined, by the 33rd draw, to the tiny figure of 0.0001354.

Or, one chance in 7,384.

The Answer

So we can now state in precise terms the probability of getting the Chu DDJ result by the operation of chance alone. It is this:

P(33W) = 0.0001354, or 1 chance in 7,384

Now, to recapitulate, a certainty of 99% (1 chance in 100 that the result could have been produced by chance) is normally regarded as decisive. A certainty of 99.9% (1 chance in 1,000) is normally regarded as decisive even in situations where extreme caution is required. The present result (1 chance in 7,384, or 99.99%) is more than seven times as certain as is required in extreme caution situations. It therefore seems safe to conclude that the chance that the Chu DDJ text was selected from a complete 81-chapter DDJ is so remote as to be impossible.

The hypothesis of an incomplete DDJ text behind the Chu selection is thus decisively confirmed, and the accretional model of that text is supported beyond any of its rivals.

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