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

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.

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.

Let A,B be two events. We say that A and B are independent of each other iff:

• $\mathbb{P}(A \cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B)$

Now, note that A and B are independent of each other if and only if $\mathbb{P}(A|B) = \mathbb{P}(A)$. In other words, A and B are independent of each other if and only if the realization of one of the events does not affect the conditional probability of the other. Assume that we perform two random experiments independent of each other, meaning that the two experiments do not interact. That is, the experiments have no in influence on each other. Let A denote an event related to the first experiment, and let B denote an event related to the second experiment. We can see that in this situation the equation $\mathbb{P}(A \cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B)$ must hold in order for it to be independent. Thus, we have that A and B are independent if and only if it satisfies the above definition.

There are also many other cases where events related to a same experiment are independent, in the sense of the above definition. For example for a fair die, the events A = {1,2} and B = {2, 4, 6} are independent. Lets also say if two disjoint events intersection/union probability is equal to zero, in this case you will know they are dependent of one another because then if one event occurs the other doesn't hence dependency. There can also be more than two independent events at a time.

Here read this, hopefully it will make more sense http://en.wikipedia.org/wiki/Conditional_probability.