Why is intersection of two independent set probability a multiplication process? Why is the probability of intersection of two independent sets $A$ and $B$, a multiplication of their respective probabilities i.e. Why is
$$\mathbb{P}(A \cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B)?$$
this question is about the intuition behind the definition of independence of sets in a probability space

Perhaps, one way of looking at it is the fact that intersection means to add conditions. In that sense, for every individual satisfying the condition A you have to compute how many satisfy condition B, thus the multiplication issue. Note that satisfying two conditions is scarcer than satisfying just one, so the "weight" of individuals satisfying both conditions respect the whole population must be less than the weight of the ones satisfying just one condition. Then, given the fact that probabilities are less than or equal one, multiplication seems a reasonable way of describing the process.
I know it is not the same in a formal sense, but is like saying for every row, compute how many columns, then you get the total number of pieces.
