MS 473-474 (1903) - Lowell Lecture VII

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and in the second 350, 28 fives, which is not particularly unlikely under the supposition of a chance distribution. During the process of counting these 5's, it occurred to me that as the expression of a rational fraction in decimals takes the form of a circulating decimal in which the figures recur with perfect regularity, so in the expression of a quantity like π, it was naturally to be expected that the 5's, or any other figure, should recur with some approach to regularity. In order to find out whether anything of this kind was discernible I counted the fives in 70 successive sets of 10 successive figures each. Now were there

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no regularity at all in the recurrence of the 5's, there ought among these 70 sets of ten numbers each to be 27 that contained just one five each; and the odds against there being more than 32 of the seventy sets that contain just one five each is about 5 to 1. Now it turns out upon examination that there are 33 of the sets of ten figures which contain just one 5. It thus seems as if my surmise were right that the figures will be a little more regularly distributed than they would be if they were entirely independent of one another. But there is not much certainty about it. This will serve to illustrate what this kind of induction is like, in which the question to be decided is how far a

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given succession of occurrences are independent of one another and if they are not independent what the nature of the law of their succession is.

In the second variety of statistical induction, we are supposed to know whether the occurrences are independent or not, and if not, exactly how they are connected, and the inquiry is limited to ascertaining what the ratio of frequency is, after the effects of the law of succession have been eliminated. As a very simple example, I will take the following. The dice that are sold in the toy shops as apparatus for games that are sold are usually excessively irregular. It is no great fault, but rather enhances the

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christmas gaiety. Suppose, however, that some old frump with an insatiable appetite for statistics to get hold of a die of that sort, and he will spend his Christmas in throwing it and recording the throws in order to find out the relative frequency with which the different faces turn up. He assumes that the different throws are independent of one another and that the ten thousand or so which he makes will give the same relative frequencies of the different faces as would be found among any similar large number of throws until the die gets worn down. At least he can safely assume that this will be the case as long as the die is thrown out of the same box by the same person in the same fashion.

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