another runner in the night

14 March 2009

All these experiments, however, are thrown completely into the shade by the enormously extensive investigations of the Swiss astronomer Wolf, the earliest of which were published in 1850 and the latest in 1893. In his first set of experiments Wolf completed 1000 sets of tosses with two dice, each set continuing until every one of the 21 possible combinations had occurred at least once. This involved altogether 97,899 tosses, and he then completed a total of 100,000. These data enabled him to work out a great number of calculations, of which Czuber quotes the following, namely a proportion of .83533 of unlike pairs, as against the theoretical value .83333, i.e. 5/6. In his second set of experiments Wolf used two dice, one white and one red (in the first set the dice were indistinguishable), and complete 20,000 tosses, the details of each result being recorded in the Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich. He studied particularly the number of sequences with each die, and the relative frequency of each of the 36 possible combinations of the two dice. The sequences were somewhat fewer than they ought to have been, and the relative frequency of the different combinations very different indeed from what theory would predict. The explanation is easily found; for the records of the relative frequency of each face show that the dice must have been very irregular, the six face of the white die, for example, falling 38 percent more often than the four face of the same die. This, then, is the sole conclusion of these immensely laborious experiments,—that Wolf's dice were very ill made. Indeed the experiments could have had no bearing except upon the accuracy of his dice. But ten years later Wolf embarked upon one more series of experiments, using four distinguishable dice,—white, yellow, red, and blue,—and tossing this set of four 10,000 times. Wolf recorded altogether, therefore, in the course of his life 280,000 results of tossing individual dice. It is not clear that Wolf had any well-defined object in view in making these records, which are published in curious conjunction with various astronomical results, and they afford a wonderful example of the pure love of experiment and observation.

John Maynard Keynes
A Treatise on Probability (1921)
Part V Ch XXIX §18 ¶2 (pp 362-3)

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