Hi there

Your question is an interesting one and I am a little puzzled as to why I myself have not considered this basic question! I had to look into this so I am not 100% sure of the answer but here is my best attempt;

As with all theories we often start with axioms that are simply stated or defined and accepted. All other theories derive from these.

If the sample space has elements, i.e A,B C .. etc where A can contain one or more elementary outcomes then the classical axioms of probability according to Kolmogorov are;

Let A and B be any events then

when .

The definition of conditional probability

satisfies the above axioms.

We can state intuitively that if knowing B does not affect A, i.e , then from the conditional probability definition we can state that

which leads to the multiplication of independent events.

Thus as far as I can see the independent events that multiply comes directly from the definition of conditional probability. If we were to make another definition then this might change.