Abstract
There are many different types of error; one can describe where the probability calculus best describes error. Calculable random error is only one type. Bessel describes the many sources of error when using an astronomical instrument. The probability calculus may be applied when total random error is compounded from many sources, and where each makes a small independent contribution. The situation is well described by Gauss’s law of error, but that law is not absolute. There are major theoretical differences between errors in observation and games of chance, for example. In general one cannot draw a strict parallel. The probability calculus is not well applied in the social sciences, and in medical statistics. The statistical treatment of majority decisions is also discussed, as in decisions made by juries. A critique is offered of the account of them given by Laplace and by Poisson.
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Notes
- 1.
Here as in what follows, it is understood that this is always a definite amount with a small and definite range. Strictly speaking this concerns the probability that the error is larger than a specific value x, but smaller than x + Δx .
- 2.
Bessel, Gesammelte Abhandlungen, vol. 2, p. 388. [Bessel, F.W. (1876). Gesammelte Abhandlungen, Band 1.: Theorie der Instrumente, Stellarastronomie, Mathematik. Leipzig: W. Engelmann.]
- 3.
Cf. for example, Hagen, Grundzüge der Wahrscheinlichkeits-Rechnung, p. 54. [Hagen, G. (1867). Grundzüge der Wahrscheinlichkeitsrechnung. 2d. ed. Berlin: Ernst & Korn.; Gauss, C.F. (1821). Theoria combinationis observationum erroribus minimus obnoxiae, pars prior. Göttingen: Apud Henricum Dieterich, 6–7. Also in: Gauss, C.F. (1880). Werke 4. Göttingen: Königlichen Gesellschaft der Wissenschaften.]
- 4.
Op. cit. p. 31.
- 5.
Lexis, W. (1877). Zur Theorie der Massenerscheinungen in der menschlichen Gesellschaft. Freiburg i. B.: Fr. Wagner.
- 6.
Loc. cit. pp. 64 and 91.
- 7.
Loc. cit. p. 83.
- 8.
Loc. cit. p. 91.
- 9.
Among others, cf. Drobisch (1880). Ueber die nach der Wahrscheinlichkeits-Rechnung zu erwartende Dauer der Ehen. Berichte über die Verhandlungen der königlich – sächsischen Gesellschaft der Wissenschaften zu Leipzig, mathematische – physische Klasse, 32, 1–21. “Let us indicate the number of men who survive at the end of each year m, m + 1, m + 2 … as being lm, lm + 1, lm + 2 …; these are numbers to be taken from mortality tables. Then the probabilities that a man of m years of age will still be alive after 1, 2, 3 … etc. years will be: \( \frac{l_{m+1}}{l_m} \), \( \frac{l_{m+2}}{l_m} \), \( \frac{l_{m+3}}{l_m} \), etc.. Now if lw, lw + 1, lw + 2 … signify the same numbers for women, then similarly the probabilities that a woman of w years of age will still be alive after 1, 2, 3 … etc. years will be: \( \frac{l_{w+1}}{l_w} \), \( \frac{l_{w+2}}{l_w} \), \( \frac{l_{w+3}}{l_w} \), etc.. The product of these probabilities: \( \frac{l_{m+1}\bullet {l}_{w+1}}{l_m\bullet {l}_w} \), \( \frac{l_{m+2}\bullet {l}_{w+2}}{l_m\bullet {l}_w} \), \( \frac{l_{m+3}\bullet {l}_{w+3}}{l_m\bullet {l}_w} \), etc. are the probabilities that this man and this woman may live together or jointly after 1, 2,or 3 years, and the sum of these probabilities gives the number of years they may hope to live together, to the end. Then in that much, this number may be termed the duration of the relationship, or if the relation is marital, as the length of the marriage.”
- 10.
Fismer, F.H. (1873). Die Resultate der Kaltwasserbehandlung bei der acuten croupösen Pneumonie im Basel Spitale von Mitte 1867 bis Mitte 1871. Deutsches Archiv für klinische Medizin. 11, pp. 395 & 396.
- 11.
In the face of these circumstances, if one believes one must require that conclusions should be based solely on the observation and the comparison of cases which are really entirely of the same kind, then I think that means one requires the subject of the investigation to be very different than it is in reality.
- 12.
Fick, Medicinische Physik, 3d. ed., p. 430. [Fick, A. (1885). Die medicinische Physik. 3d. ed. Braunschweig: Friedrich Vieweg und Sohn.]
- 13.
Liebermeister, C. (1877). Ueber Wahrscheinlichkeitsrechnung in Anwendung auf therapeutische Statistik, Sammlung klinischer Vorträge. (Innere Medicin No. 31 – 64), 110, 935–962.
- 14.
Here I glean from this circumstance that still, only such general conditions are considered whose constancy would determine a normal dispersion.
- 15.
Op. cit. p. 18.
- 16.
de Condorcet, M.J.A.N. (1785). Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. Paris, de l’Imprimerie Royale.
- 17.
Laplace (1812). Théorie analytique des probabilités. 3d. ed., pp. XXXVI & 460. [Laplace, P.-S. (1812). Théorie analytique des probabilités. Paris: Mme Ve Courcier.]
- 18.
Poisson, S.-D. (1837). Recherches sur la probabilité des jugements en matière criminelle et en matière civile. Paris: Bachelier, Imprimeur-Libraire.
- 19.
Op. cit. Laplace, p. 460.
- 20.
Op. cit. Poisson, p. 14.
- 21.
For the purpose of an empirical assessment it would be necessary to present particular cases in succession to different juries for their decision. Of course there are good reasons why this should not happen; one is likely to disregard how the material for such an investigation should be provided – material from which one might gain some picture of the “degree of randomness” attached to verdicts.
- 22.
Also see a similar exposition given by Laplace in his Théorie analytique …, p. LXXXVI of his Introduction. [Laplace, P.-S. (1812). Théorie analytique des probabilités. Paris: Mme Ve Courcier.]
- 23.
« Pour les jurés du ressort de chaque cour d’assises et pour chacun des deux genres de crimes que nous avons distingués on doit donc concevoir qu’il y a une certaine probabilité z, jugée suffisante et nécessaire pour la condamnation. Cela étant, la chance u, qu’un juré pris au hazard sur la liste de ce département ne se trompera pas dans son vote, est la probabilité qu’il jugera celle (la probabilité) de la culpabilité de l’accusé égale ou supérieure à z si elle l’est effectivement, ou bien, inférieure à z, si en effect, elle n’atteint pas cette limite. »
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Niall, K.K. (2023). More Applications of the Probability Calculus. In: Johannes von Kries: Principles of the Probability Calculus. Studies in History and Philosophy of Science, vol 59. Springer, Cham. https://doi.org/10.1007/978-3-031-36506-5_9
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