What Is The Theorem Of Compound Probability?

The probability of occurrence of the event A as well as B is given by the product of (unconditional) probability of A and conditional probability of B, assuming that A has actually occurred, i.e.

Probability of (A and B) = Probability of A X Conditional Probability of B, assuming A

In symbolic form, P(AB) = P(A).P(B|A)

This is also known as Multiplication Theorem.

Proof: Suppose that a random experiment has n mutually exclusive, exhaustive and equally likely outcomes, among which m cases are favourable to an event A. So, the unconditional probability of A is

P(A) = m / n

Out of these m cases, let m1 cases be favourable to another event B also; i.e. the number of cases favourable to A as well as B is m1. Hence, by the classical definition of probability,

P(AB) = m1 / n

It may be noticed that once event A is known to have actually occurred, the occurrence of B as well is limited to only m1 cases out of m (in which A occurs). So, the conditional probability of B, assuming that A has already occurred, is

P(B|A) = m1 / m

We find that, m1/n = (m/n).(m1/m)

i.e. P(AB) = P(A).P(B|A)