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In mathematics terminology, a function is defined over two sets. One is is input set and other one is output set and for a particular input element, we the following notation

$$f(x) = y$$

where $x$ is the input element and $y$ is the output element.

Coming to conditional probability the notation is as follows

$$p(A/B) = \dfrac{p(A\cap B)}{p(B)}$$ The numerator and denominator are perfectly okay as per notations. But coming to the notation for conditional probability, the input is just $A$ under the assumption of $B$ occurrence.

What is the reason behind that notation, since $A/B$ is not an eligible input where $A$ has to be?

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  • $\begingroup$ Also common is the vertical line $p(A | B)$. So, as explained in Mauro's answer, it is a way to write a function of two arguments $A$ and $B$. $\endgroup$ – Gerald Edgar Jan 24 '18 at 11:57
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See: A.N. Kolmogorov, Foundations of the Theory of Probability, (1956: original ed.1933), page 3:

To each set $A$ in [the collection of elementary events] $\mathfrak F$ is assigned a non-negative real number $\mathsf P(A)$.

In symbols:

$\mathsf P(A): \mathfrak F \to \mathbb R$.

And see page 6:

If $\mathsf P(B) > 0$, then the quotient $\mathsf P_B(A)=\dfrac {\mathsf P(AB)}{\mathsf P(B)}$ is defined to be the conditional probability of the event $A$ under the condition $B$.

Formally the conditional probability is a function of two variables:

$\mathsf P_B(A): \mathfrak F \times \mathfrak F \to \mathbb R$.

In other terms, for each $B$, we have that:

$\mathsf P_B(A): \mathfrak F \to \mathbb R$.

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