Unizor - Probability - Conditional Probability Definition

Assume we have a set of N elements that represents a sample space of N elementary events with equal chances of occurrence and, therefore, a probability measure of 1/N allocated to each element of this set. Assume further that we are interested in an event represented by a subset A of this set that contains M elements. The condition we impose on this model is the fact that only K elementary events represented by elements of a subset B can actually occur as a result of an experiment. This forces us to redistribute the probability measure to only elements of this subset B, allocating 1/K to each of them and 0 to all other elements outside of a subset B. Since we are interested in the elements of a subset A, we have to choose from all M of them only those that are also a part of a subset A, that is only those from the intersection A∩B, because all other elements have a measure of 0 allocated to them. Let's assume that A∩B contains L elements. Then the conditional probability of the event represented by a subset A under condition represented by a subset B equals to P(A|B)=L/K. But exactly the same result can be obtained by dividing L/N by K/N . Notice now that L/N is P(A∩B) and K/N is P(B). Therefore, we have shown that
P(A|B) = P(A∩B) / P(B)

This equation basically means that a conditional probability of some random event A under a condition of an occurrence of another random event B is a fraction of a measure allocated to the occurred random event B taken by elementary events A∩B common between this condition B and a random event A we are interested in. When events are graphically represented as figures on a plane and probability is interpreted as the area, this equation becomes quite obvious.

This simple equation can be extended to cases of non-equal chances of elementary events and also to infinite sample spaces. Actually, in a rigorously constructed Probability Theory this equation is used as a definition of the conditional probability.