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During the late 16th century and early 17th century, published work about probability theory (e.g. Liber de ludo aleae by J. Cardan published in 1663 but writen around 1564) studied dice games using equiprobability on a finite sample space.

Nowadays, we use terms such as equiprobability and uniform distribution to refer to their work, but at the time those people didn't use these terms.

Who coined the terms uniform and equiprobability ?

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  • $\begingroup$ As often in similar cases, we may have an issue about translation: when yiou assert that Cardan uses "equiprobability", you must take into account that the Liber de ludo aleae was written in Latin. Thus, which is the latin term used by Cardan ? $\endgroup$
    – Mauro ALLEGRANZA
    May 18 '18 at 11:28
  • $\begingroup$ @MauroALLEGRANZA Ah! you're right, I didn't think about these translation issues. I expect a variant of "equiprobability" could have been used in early probability theory (e.g. a variant was definitely used by Laplace, maybe by Cardan too) but I expect the term "uniform" to be much more recent and linked with the work of Borel on continuous space (think about the continuous uniform probability). My expectation is that by that time, translation is not an issue anymore. But all this is speculative. $\endgroup$
    – Julien__
    May 18 '18 at 12:36
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    $\begingroup$ Jeff Miller's site gives the first use of "uniform distribution" as attributed to J.V. Uspensky in his 1937 text Introduction to Mathematical Probability. The same site entry on equiprobable gives its first use dated 1853 in a review of Essays on Life Assurance by Edward Sang printed in a review appearing the an actuarial journal. $\endgroup$
    – NWR
    May 18 '18 at 15:57
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The first uses of what we call "uniform distribution" occur very early, discrete arguably already in Cardano, and continuous in Simpson and Bayes. According to Handbook of Beta Distributions:

"One of the first records that mentions the continuos uniform distribution is the famous paper by the reverend Thomas Bayes (1763) (only a few years after Simpons' written records in 1757)."

The paper is Essay towards solving a Problem in the Doctrine of Chances, where conditional probabilities and the Bayes's formula are introduced. Bayes considered independent experiments with the probability of success being some number $p$ between $0$ and $1$ and made the number itself to be random with the probability to fall into any interval between $0$ and $1$ equal to the length of that interval. However, the term "uniform distribution" itself is much more recent. According to James Landau's entry to Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics:

"Uniform distribution appears in 1937 in Introduction to Mathematical Probability by J. V. Uspensky. Page 237 reads, "A stochastic variable is said to have uniform distribution of probability if probabilities attached to two equal intervals are equal." This is a slight variant of the modern terminology, which would be "a variable is said to be uniformly distributed" or "a variable from the uniform distribution". Uniformly distributed is found in H. Sakamoto, "On the distributions of the product and the quotient of the independent and uniformly distributed random variables," Tohoku Math. J. 49 (1943)."

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Regarding English, I think that the first treatise was Abraham De Moivre's treatise The Doctrine of Chances (1718).

For the discrete case, see page 7 :

"If the Events in question are $n$ in number and are such as have the same number $a$ of Chances by whcih they may Happen..."

And see also William Emerson (1701 – 1782)'s The Laws of Chance (1776), page 3 :

"AXIOM I. In computing the number of Chances, it is supposed that all Chances are equal, or made with equal facility."

And page 4 :

"SCHOLIUM. As $1$ represents a certainty, or when an event has an infinitely great probability of happening, so $\frac 1 2$ represents an equal probability [emphasis added] for happening or failing. For $1 − \frac 1 2 = \frac 1 2$."

For a good overview of the early history of probability in general and of equipossibility in particular, see Ian Hacking, The Emergence of Probability : A Philosophical Study, Cambridge UP (2nd ed 2006), Ch.14, page 122-on.

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  • $\begingroup$ Thank you for your interest in my question, Mauro (+1). Anything about "uniform" more specifically? $\endgroup$
    – Julien__
    May 18 '18 at 13:42

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