P(A and B)/P(B). This may be written:
P(A|B)P(B) = P(A and B)
Also you do not need to assume that A and B are independent.
Suppose we are talking about the result of throwing a single die, and let
A be the event that an even number is thrown, and B be the event that
a number greater than 2 is thrown. Then P(A)=1/2, P(B)=4/6, P(A and B)=2/6
and P(A|B) is clearly 2/4 (there are 4 possible faces which show given that
B has occured of which 2 are even), and the formula tells us that:
P(A|B)=P(A and B)/P(B)=(2/6)(6/4)=2/4.
This is essentialy Bayes theorem, since:
P(A and B)=P(A|B)P(B)=P(B|A)P(A),