Mémoire Sur Les Probabilités∗

I intend to treat in this Memoir two important points in the analysis of chances which do not seem yet to have been sufficiently deeply studied: the first has for object the manner of calculating the probability of events composed of simple events of which one does not know the respective probabilities; the object of the second is the influence of past events on the probability of future events, and the law according to which, in its expansion, shows us the causes which have produced them. These two objects, which have much analogy between them, depend upon a very delicate metaphysics, and the solution of the problems which are relative to them require some new artifices of analysis; they form a new branch of the theory of probabilities, of which the usage is indispensable when we wish to apply this theory to civil life. I give, relative to the first, a general method to determine the probability of any event, when we know only the law of possibility of the simple events, and, in the case where this law is unknown, I determine that of which we must make usage. The consideration of the second object leads me to speak of births: as this matter is one of the most interesting in which we are able to apply the Calculus of probabilities, I manage so to treat with all care owing to its importance, by determining what is, in this case, the influence of the observed events on those which must take place, and how, by its multiplying, they uncover for us the true ratio of the possibilities of the births of a boy and of a girl. By generalizing next these researches, I arrive to a method to determine, not only the possibilities of simple events, but yet the probability of any future event, when the observed event is very compound, whatever be moreover its nature. I give, on this occasion, the solution of some interesting problems in the natural history of man, such as the one of the more or less facility of the births of boys relatively to those of the girls in different climates: it is here chiefly that it is necessary to have a rigorous method in order to distinguish, among the observed phenomenon, those which are able to depend on chance, from those which depend on the particular causes, and in order to determine with what probability these last indicate the existence of these causes. The principal difficulty that we encounter in these researches depends on the integration of certain differential functions which