Editing Independence (probability theory) (section)

== History ==
Before 1933, independence, in probability theory, was defined in a verbal manner. For example, [[de Moivre]] gave the following definition: “Two events are independent, when they have no connexion one with the other, and that the happening of one neither forwards nor obstructs the happening of the other”.<ref>Cited according to: Grinstead and Snell’s Introduction to Probability. In: The CHANCE Project. Version of July 4, 2006.</ref> If there are n independent events, the probability of the event, that all of them happen was computed as the product of the probabilities of these n events. Apparently, there was the conviction, that  this formula was a consequence of the above definition. (Sometimes this was called the Multiplication Theorem.), Of course, a proof of his assertion cannot work without further more formal tacit assumptions.

The definition of independence, given in this article, became the standard definition (now used in all books) after it appeared in 1933 as part of Kolmogorov's axiomatization of probability.<ref>[[Andrey Kolmogorov|Kolmogorov, Andrey]] (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung (in German). Berlin: Julius SpringerTranslation: Kolmogorov, Andrey (1956). Translation:Foundations of the Theory of Probability (2nd ed.). New York: Chelsea. ISBN 978-0-8284-0023-7.</ref> [[Andrey Kolmogorov|Kolmogorov]] credited it to [[Sergei Bernstein|S.N. Bernstein]], and quoted a publication which had appeared in Russian in 1927.<ref>[[Sergei Bernstein|S.N. Bernstein]], Probability Theory (Russian), Moscow, 1927 (4 editions, latest 1946)</ref>

Unfortunately, both Bernstein and Kolmogorov had not been aware of the work of the [[Georg Bohlmann]]. Bohlmann had given the same definition for two events in 1901<ref>[[Georg Bohlmann]]: Lebensversicherungsmathematik, Encyklop¨adie der mathematischen Wissenschaften, Bd I, Teil 2, Artikel I D 4b (1901), 852–917</ref> and for n events in 1908<ref>[[Georg Bohlmann]]: Die Grundbegriffe der Wahrscheinlichkeitsrechnung in ihrer Anwendung auf die Lebensversichrung, Atti del IV. Congr. Int. dei Matem. Rom, Bd. III (1908), 244–278.</ref> In the latter paper, he studied his notion in detail. For example, he gave the first example showing that pairwise  independence does not imply mutual independence.
Even today, Bohlmann is rarely quoted. More about his work can be found in ''On the contributions of Georg Bohlmann to probability theory'' from [[:de:Ulrich Krengel]].<ref>[[:de:Ulrich Krengel]]: On the contributions of Georg Bohlmann to probability theory (PDF; 6,4 MB), Electronic Journal for History of Probability and Statistics, 2011.</ref>