Why is intersection of two independent set probability a multiplication process?

Solution 1:

The fact that the probability of the intersection of independent events $A$ and $B$ is the product of their probabilities is actually the definition of independent events.

Solution 2:

If

• half the slices of a pizza have anchovies ($P(A)=\frac12$), and
• you take a third of the slices of pizza ($P(B)=\frac13$) independently of whether they have anchovies, then
• the anchovy slices that you have are one-sixth of all the slices of pizza ($P(A\cap B) = \frac16$).

This is because if your taking of slices is truly independent of their having anchovies, then

• you will take a third of the anchovy slices ($P(A\cap B) = \frac13 P(A)$) and a third of the non-anchovy slices;

• equivalently, half the slices you have will have anchovies ($P(A\cap B) = \frac12 P(B)$) and half will not.