Rules of Probability

There are two important rules for computation of probabilities which are otherwise known as the ‘Theorems of Probability’. They are:

(1)  Rule of Multiplication, or Multiplication Theorem, and

(2)  Rule of Summation, or Addition Theorem.

(3)  Rule of Multiplication. According to this rule, the probabilities of two, or more related events are multiplied with each other to find out the net probability of their joint occurrence.

This rule of multiplication is applied to the problems of compound events where:

(a)  The related events are independent of each other, and

(b)  The related events are not mutually exclusive.

In such cases the net probability of the several related events say A, B and C is calculated as under:

P (AB) = P (A) × P (B)

And     P (ABC) = P(A) × P (B) × P (C)

(2) Rule of Summation. According to this rule, the probabilities of two or more related events will be summed up to find out the probability of their joint occurrence.

This rule of summation is applied to the problems of compound events where.

(a)  The related events are dependent of each other, and

(b)  The related events are mutually exclusive.

The rule of summation again comprises of two sub rules, viz.

(i)  Rule of pure addition

(ii)  Rule of both additions and subtraction.

(i) Rule of Pure addition. According to this rule the probabilities of all the related events are added together to compute the probability of their joint occurrence. This rule is applied only, where all the related events are mutually exclusive of each other (i.e., when one takes place, the other cannot), and there is no intersection (one’s happening with the happening of another) between any two, or more related events.

In such a case, the probability of the compound events is computed as under:

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

Or      P (A ∪  B) = P (A) + P (B) – P (A ∩ B)

In the case of three events the formula is modified as under:

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

-P(A ∩ C) – P(BC) + P(A ∩ B ∩ C)