Formula for Conditional probability doesn't make sense to me conditional probability is defined us:
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
But if (and there is the part where i might be missing something) $P(A \cap B) = P(A) \cdot P(B)$ then:
$$ \frac{P(A \cap B)}{P(B)} = \frac{P(A) \cdot P(B)}{P(B)} = P(A) $$ 
which doesn't make any sense. Why would the formula exist in the first place. I'm confused.
Thanks for helping me :).
 A: The easiest way to picture conditional probability is by viewing a Venn diagram.  (Sorry, I am doing this quickly and have not learned to generate one in MathJAX yet.)  In a simple diagram with two sets A and B, think of conditional probability as the ratio of the area of the intersection of the two circles to the area of the "given" circle (for A given B, that circle is B).  
Imagine redefining the universe to be the given set (B in the previous example); the conditional probability is then the probability of A in that universe. 
A: Here's a more intuitive understanding that leads to the definition. (Sorry I'm not very good at latex.)
Take two events A and B, where they aren't necessarily independent. We seek to find the probability that both A and B occur. This turns out to be P(A and B).
Now, let's say A happens first. Therefore, P(A and B) = P(A)("something"). But since P(B|A) is defined as the probability of B occurring given A has already occurred, it is intuitively true that P(A and B) = P(A)P(B|A). Dividing P(A) on both sides yields the desired formula. 
Note that when the A and B are independent events, P(B|A) = P(B), because the events don't affect each others' outcomes. This simplifies to the regular formula for probabilities of independent events.
