The inverse application of the rules of probability entirely depends upon a proposition which may be thus stated, nearly in the words of Laplace.[151] *If an event can be produced by any one of a certain number of different causes, all equally probable à priori, the probabilities of the existence of these causes as inferred from the event, are proportional to the probabilities of the event as derived from these causes.* In other words, the most probable cause of an event which has happened is that which would most probably lead to the event supposing the cause to exist; but all other possible causes are also to be taken into account with probabilities proportional to the probability that the event would happen if the cause existed. Suppose, to fix our ideas clearly, that E is the event, and C_{1} C_{2} C_{3} are the three only conceivable causes. If C_{1} exist, the probability is *p*_{1} that E would follow; if C_{2} or C_{3} exist, the like probabilities are respectively *p*_{2} and *p*_{3}. Then as *p*_{1} is to *p*_{2}, so is the probability of C_{1} being the actual cause to the probability of C_{2} being it; and, similarly, as *p*_{2} is to *p*_{3}, so is the probability of C_{2} being the actual cause to the probability of C_{3} being it. By a simple mathematical process we arrive at the conclusion that the actual probability of C_{1} being the cause is *p*_{1}/(*p*_{1} + *p*_{2} + *p*_{3}); [151] *Mémoires par divers Savans*, tom. vi.; quoted by Todhunter in his *History of the Theory of Probability*, p. 458. and the similar probabilities of the existence of C_{2} and C_{3} are, *p*_{2}/(*p*_{1} + *p*_{2} + *p*_{3}) and *p*_{3}/(*p*_{1} + *p*_{2} + *p*_{3}). The sum of these three fractions amounts to unity, which correctly expresses the certainty that one cause or other must be in operation. We may thus state the result in general language. *If it is certain that one or other of the supposed causes exists, the probability that any one does exist is the probability that if it exists the event happens, divided by the sum of all the similar probabilities.* There may seem to be an intricacy in this subject which may prove distasteful to some readers; but this intricacy is essential to the subject in hand. No one can possibly understand the principles of inductive reasoning, unless he will take the trouble to master the meaning of this rule, by which we recede from an event to the probability of each of its possible causes.