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In the 1738 second edition of The Doctrine of Chances, de Moivre writes,

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.

Two Events are dependent, when they are so connected together as that the Probability of either's happening is altered by the happening of the other.

and

The Probability of the happening of two Events dependent, is the product of the Probability of the happening of one of them, by the Probability which the other will have of happening, when the first shall be consider'd as having happened; and the same Rule will extend to the happening of as many Events as may be assigned.

Is this the earliest use of the concepts of independence/dependence and conditional probability? Whether or not it is, are there other early discussions I should be aware of? In particular, are these concepts always explained as de Moivre explains them? And is conditional probability always regarded as simply a tool for calculating joint probabilities? When does this change?

As a side question, a couple paragraphs later, de Moivre writes,

We had seen before how to determine the Probability of the happening or failing of as many Events independent as may be assigned; we have seen likewise in the preceding Article how to determine the Probability of the happening of as many Events dependent as may be assigned: but in the case of Events dependent, how to determine the Probability of the happening of some of them, and at the same time the Probability of the failing of some others, is a disquisition of a greater degree of difficulty; which for that reason will be more conveniently transferred to another place.

This is a curious remark, for we moderns naturally regard the failing of an event to happen as nothing other than the happening of the complement event. Can someone explain why de Moivre regards finding joint probabilities of the happenings and failings of events as harder than finding joint probabilities of the happenings of events? Does he, as he promises, discuss the former problem in some other place?

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The earliest discussion of conditional probabilities goes back to the analysis of Pascal and Fermat (1654) of the problem of points: given that team $A$ has won $m$ games and team $B$ has won $n$ games, what is the probability that $A$ will win the series? A little later (1665), Christiaan Huygens and John Hudde corresponded on the difference between conditional and unconditional probabilities, as reprinted here and as discussed by I. Hacking in The Emergence of Probability.

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  • $\begingroup$ Thanks, Carlo. I am familiar with the problem of points, and I have just reviewed chapter 11 of Hacking's book on the the Huygens/Hudde correspondence (having checked, in vain, the index of that book for "independence" and "conditional probability" before even posting). I guess it is a subtle question when the concepts of independence and conditional probability emerge, and the emergence might be gradual, but I would not be inclined to say that they have emerged in either the Pascal/Fermat discussion or the Huygens/Hudde discussion. They don't have the P(A,B)=P(B)P(A|B) formula, do they? $\endgroup$
    – user48301
    Commented Apr 17, 2014 at 16:53
  • $\begingroup$ they certainly do not; the notation itself $P(A|B)$ is from much later $\endgroup$
    – Carlo Beenakker
    Commented Apr 17, 2014 at 18:56
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    $\begingroup$ Of course the notation is from much later, but I was just talking about the formula. De Moivre doesn't have the notation, but he introduces independence/dependence and conditional probability almost exactly as it is usually introduced to beginning students today, and that includes the P(A,B)=P(B)P(A|B) formula (see the second passage I quoted in the question). $\endgroup$
    – user48301
    Commented Apr 17, 2014 at 19:42

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