Multiplication TheoremTheorem: If A and B are two independent events, then the probability that both will occur is equal to the product of their individual probabilities. P(A∩B)=P(A)xP(B) Proof: Let event A can happen is n_{1}ways of which p are successful B can happen is n_{2}ways of which q are successful Now, combine the successful event of A with successful event of B. Thus, the total number of successful cases = p x q We have, total number of cases = n_{1} x n_{2}. Therefore, from definition of probability P (A and B) =P(A∩B)= We have P(A) =,P(B)= So, P(A∩B)=P(A)xP(B) If, there are three independent events A, B and C, then P(A∩B∩C)=P((A∩B)∩C)= P(A∩B)xP(C) =P(A) x P(B) x P(C). 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) = P(B) =P(a red ball) = By Multiplication Theorem P(A) and P(B) = P(A) x P(B) =
