Theorem of Probability

There are two important theorems of probability, namely:

(i)    The Additions Theorem; and

(ii)    The Multiplication Theorem.

Additions Theorem

The additions theorem states that 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. Symbolically,

            P (A or B) = P (A) + P (B)

Proof of the Theorem.

If an event A can happen in a₁ ways and B in a₂ ways, then the number of ways in which either event can happens is a₁ + a₂. If the total number of possibilities is n, then by definition the probability of either the first or the second event happening is

            a₁ + a₂ = a₁ + a₂
               _n          n       n

But    a1 = P(A)
   ___ n

and     a2 = P(B)
    ____n

Hence P (A or B) = P(A) + P (B)

The theorem can be extended to three or more mutually exclusive events. Thus,

    P (A or B or C) = P (A) + P (B) + P (C)

Multiplication Theorem

This theorem states that if two events A and B are independent the probability that they both will occur is equal to the product of their individual probabilities. Symbolically, if A and B are independent, then

            P (A and B) = P(A) X P(B)

The theorem can be extended to three or more independent events. Thus
            P (A and B) = P(A) X P(B) X P(C)

Proof of the Theorem.

If an event can happen in n₁ ways of which a₁ are successful and the event B can happen in n₂ ways of which a₂ are successful, we can combine each successful event in the first with each successful event in the second case. Thus the total number of successful happenings in both cases is a₁Xa₂. Similarly, the total number of possible cases is n₁ X n₂.

Then by definition the probability of the occurrence of both events

            a₁ x a₂ = a₁ x a₂
            n1 x n2 = n1   n2
But     a₁ = P (A)
    ___n₁
and     a₂ = P (B)
    ___n2

Hence, P (A and B) = P (A) X P (B).

In a similar way the theorem can be extended to three or more events.

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