Conditional Probability

The conditional probability of an event  assuming that  has occurred, denoted , equals

(1)

which can be proven directly using a Venn diagram. Multiplying through, this becomes

(2)

which can be generalized to

(3)

Rearranging (1) gives

(4)

Solving (4) for  and plugging in to (1) gives

(5)

REFERENCES:

Papoulis, A. "Conditional Probabilities and Independent Sets." §2-3 in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 33-45, 1984.