Some pondering leads me to the question below, which prevents me from the reckless calculation of conditional probability...
As defined, conditional probability is:
$$ P(A|B)=P(A\cap B)/P(B). $$
So, we can see conditional probability is the ratio of 2 probabilities. If we consider conditional probability also as probability, we are literally saying some quantity describes the similar thing as its ratio. This is very bizarre because when we measure length, we can say something is 2 meters long and the other thing is 5 meters long. But we cannot consider 2/5 or 5/2 as the same thing as 2 or 5 because the ratio 2/5 or 5/2 is just another level of comparison as I understand， while the 2 or 5 is merely the comparison to the unit.
So why do we still treat conditional probability just as the ordinary probability?
I wish someone could shed some light on this thing. Or is there any other examples like this besides in probability theory?
(Thanks for so many responses.)
Almost all of the responses so far try to convince me that probability is a ratio itself. And both ratio and ratios' ratio are ratio.
Personally, I haven't found any contradiction about the conditional probability definition as far as only ratio interpretation is concerned yet. And it seems ratio is the only realm where probability is
mathematically possible. So I took it that people
generalize this ratio-based theory to a much broader realm where probability issues also arise but no apparent mathematics is applicable (such as the degree-of-belief scenario.). The only thing I can find to support this for now, is the Principle of the Permanence of Equivalent Forms and the boldness of human nature for mathematical generalization.
(I will not close this question as of now and more opinions are appreciated.)