## Theorems Of Probability

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# 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 a_{1 }ways and B in a

_{2}ways, then the number of ways in which either event can happens is a

_{1}+ a

_{2}. If the total number of possibilities is n, then by definition the probability of either the first or the second event happening is

__a__=

_{1 }+ a_{2}__a__+

_{1}__a__

_{2}_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, thenP (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 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, 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 a

_{1}Xa

_{2}. Similarly, the total number of possible cases is n

_{1}X n

_{2}.

Then by definition the probability of the occurrence of both events

__a__=

_{1}x a_{2}__a__x

_{1}__a__

_{2}n1 x n2 = n1 n2

But

__a__= P (A)

_{1}___n

_{1}

and

__a__= P (B)

_{2}_______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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