probability of an event is the ratio of the number of cases favorable to this event to the total number of possible cases.
A simple example will show how incomplete this definition is. I throw two dice. What is the probability that one of the two at least turns up a six? Each die can turn up in six different ways; the number of possible cases is 6 × 6 = 36; the number of favorable cases is 11; the probability is 11/36.
That is the correct solution. But could I not just as well say: The points which turn up on the two dice can form 6 × 7/2 = 21 different combinations? Among these combinations 6 are favorable; the probability is 6/21.
Now why is the first method of enumerating the possible cases more legitimate than the second? In any case it is not our definition that tells us.
We are therefore obliged to complete this definition by saying: '... to the total number of possible cases provided these cases are equally probable.' So, therefore, we are reduced to defining the probable by the probable.
How can we know that two possible cases are equally probable? Will it be by a convention? If we place at the beginning of each problem an explicit convention, well and good. We shall then have nothing to do but apply the rules of arithmetic and of algebra, and we shall complete our calculation without our result leaving room for doubt. But if we wish to make the slightest application of this result, we must prove our convention was legitimate, and we shall find ourselves in the presence of the very difficulty we thought to escape.
Will it be said that good sense suffices to show us what convention should be adopted? Alas!