Is there a non-circular explanation for computing the probability of the intersection of two dependent events? The explanation with which I'm familiar goes like this. Define the conditional probability P(A|B) as the probability of their intersections P(A and B) divided by the probability P(B). To figure out the probability of the intersection P(A and B) then, simply rearrange the terms of equation algebraically to obtain P(A and B) = P(A|B) X P(B).
The problem: What I'm struggling to understand is how to apply this equation in practice. It seems like whenever we want to figure out P(A and B), we need to figure out the values of P(A|B) and P(B).But to figure out the value of P(A|B), we already must figure out the value of P(A and B). So how is the equation P(A and B) = P(A|B) X P(B) helpful for figuring out the value of P(A and B)? In other words, to solve for it, we must figure out the value of P(A|B), which we can't do without knowing the value of P(A and B).

If the occurrence of an event B is affected by the occurrence of another event A then we say that A and B are dependent events. So when the experiment is performed, it is known that event A has occurred. Does this affect the probability of B? This probability is the conditional probability. In principle you need to know both P(A and B) and P(B) to figure of P(A|B) (or P(A|B) and P(B) to find P(A and B)). 
However, if the probability of B is unaffected by the prior occurrences of A, then we say that A and B are independent or that P(B|A)=P(B), which implies that P(A and B)= P(A)P(B).
