Warring States Project Arithmetic
It is not difficult (though it is perhaps a little tedious) to see why the Chu DDJ text result is decisively in favor of the Project's accretional theory of that text. We 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
We have conjectured that the DDJ was incomplete when it was drawn on by a Chu scribe to produce the Gwodyen tomb text. That 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, confirming our conjecture.
But it is 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 statistically possible?
The standard 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, all but one of them without replacing the ball after drawing it. 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. It is still better than half; in fact, drawing a White ball on this step is obviously the more likely outcome. 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 may show the results at each step, 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 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 a little less than a 90% degree of certainty that something nonrandom is happening. 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, there would be no strong support for our hypothesis of an incomplete DDJ text.
But the Chu text is larger than this. 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 (our 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 let's continue. 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.
And 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 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 AnswerSo 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, 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 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 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.
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