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The German Wikipedia article "Bernoulli-Verteilung" ("Bernoulli distribution") claims that the term "Bernoullian trials" occurs first in the textbook "Introduction to Mathematical Probability" by J.V. Uspensky (published 1937, page 45). I have found the cited statement there:

"In this chapter we shall discuss the simplest case of series of independent trials with constant probability. They are often called "bernoullian series of trials" in honor of Jacob Bernoulli who, in his classical book, "Ars conjectandi" (1713) made a profound study of such series and was led to the discovery of one of the most important theorems in the theory of probability." -- J.V. Uspensky

For me this paragraph does not support the claim that Uspensky was the first who used the term "Bernoullian trial". This leads me to my more general question: How can a historian be sure that a certain scientist or mathematician used a phrase for the first time? And more precisely: Is it right, that Uspensky was the first mathematician who used the term "Bernoullian trial"?

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Of course you cannot be sure it is the first. Merely that it is the oldest one we have found.

Mathword has your same 1937 citation for "Bernoulli trials". It has a 1934 citation for "Bernoulli distribution".

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