This is an a posteriori probability of cause. But I could not calculate it, if a convention more or less justified did not tell me in advance what is the a priori probability for the cause A to come into play; I mean the probability of this event for some one who had not observed the effect.
The better to explain myself I go back to the example of the game of écarté mentioned above. My adversary deals for the first time and he turns up a king. What is the probability that he is a sharper? The formulas ordinarily taught give 8/9, a result evidently rather surprising. If we look at it closer, we see that the calculation is made as if, before sitting down at the table, I had considered that there was one chance in two that my adversary was not honest. An absurd hypothesis, because in that case I should have certainly not played with him, and this explains the absurdity of the conclusion.
The convention about the a priori probability was unjustified, and that is why the calculation of the a posteriori probability led me to an inadmissible result. We see the importance of this preliminary convention. I shall even add that if none were made, the problem of the a posteriori probability would have no meaning. It must always be made either explicitly or tacitly.
Pass to an example of a more scientific character. I wish to determine an experimental law. This law, when I know it, can be represented by a curve. I make a certain number of isolated observations; each of these will be represented by a point. When I have obtained these different points, I draw a curve between them, striving to pass as near to them as possible and yet preserve for my curve a regular form, without angular points, or inflections too accentuated, or brusque variation of the radius of curvature. This curve will represent for me the probable law, and I assume not only that it will tell me the values of the function intermediate between those which have been observed, but also that it will give me the observed values themselves more exactly than direct observation. This is why I make it pass near the points, and not through the points themselves.