*62*

# Multiplication Theorem

**Theorem:** If A and B are two independent events, then the probability that both will occur is equal to the product of their individual probabilities.

Â Â Â Â Â Â P(Aâˆ©B)=P(A)xP(B)

**Proof:** Let event

Â Â Â Â Â A can happen is n_{1}ways of which p are successful

Â Â Â Â Â B can happen is n_{2}ways of which q are successful

Â Â Â Â Â Now, combine the successful event of A with successful event of B.

Â Â Â Â Â Thus, the total number of successful cases = p x q

Â Â Â Â Â We have, total number of cases = n_{1} x n_{2}.

Â Â Â Â Â Therefore, from definition of probability

Â Â Â Â Â P (A and B) =P(Aâˆ©B)=

Â Â Â Â Â We haveP(A) =,P(B)=

Â Â Â Â Â So, Â Â P(Aâˆ©B)=P(A)xP(B)

Â Â Â Â Â If, there are three independent events A, B and C, then

Â Â Â Â Â P(Aâˆ©Bâˆ©C)=P((Aâˆ©B)âˆ©C)= P(Aâˆ©B)xP(C)

Â Â Â Â Â Â Â Â Â Â Â Â Â Â =P(A) x P(B) x P(C).

Â Â Â Â Â In general, if there are n independent events, then

**Example:** A bag contains 5 green and 7 red balls. Two balls are drawn. Find the probability that one is green and the other is red.

**Solution:** P(A) =P(a green ball) =

Â Â Â Â Â Â Â Â P(B) =P(a red ball) =

Â Â Â By Multiplication Theorem

Â Â Â P(A) and P(B) = P(A) x P(B) =