Theorems of Probability A Multiplication Theorem If two

Theorems of Probability • A. Multiplication Theorem • If two events A and B are independent, the probability that they both will occur is equal to the product of their individual probability. If two events A and B are independent, then P (A and B) = P(A) × P(B).

A. Multiplication Theorem Proof: • If an event A can happen in n 1 ways of which a 1 are successful and • the event B can happen in n 2 ways of which a 2 are successful. • The total number of successful happenings in both cases is a 1× a 2. • The total number of possible cases is n 1× n 2. • The probability of the occurrence of both event is a 1× a 2/ n 1× n 2 = a 1/ n 1× a 2/ n 2. • But a 1/ n 1= P(A) and a 2/ n 2 = P(B). • Therefore P (A and B) = P(A) × P(B).

B. Addition Theorem: • If two events A and B are mutually exclusive, the probability of the occurrence of either A or B is the sum of the individual probability of A and B. P (A or B) = P (A) + P(B).

Proof: • If an event A can happen a 1 ways and B in a 2 ways, then the number of ways in which either event can happen is a 1 +a 2. If the total number of possibilities is n, then the probability of either the first or the second event happening is a 1 +a 2 / n= a 1/n + a 2/n. But a 1/n = P (A) and a 2 / n = P(B). • Hence P (A or B) = P (A) + P(B). • This theorem can be extended to three or more mutually exclusive events. Thus • P (A or B or C) = P (A) + P(B) + P(C).