# Dependent and Independent Events in Probability

Here, you will learn dependent and independent events in probability with examples.

Let’s begin –

## Dependent and Independent Events Formula

Two Events A and B are said to be independent if occurrence or non-occurrence of one does not affect the probability of the occurrence or non-occurrence of other.

If the occurrence of one event affects the probability of the occurrence of the other event then the events are said to be Dependent or Contigent.

For two independent events A and B :

P($$A\cap B$$) = P(A).P(B)

Often this is taken as the definition of independent events.

Note :

(a)  If A & B are independent events, then

(i)  P($$A^c\cap B^c$$) = P($$A^c$$).P($$B^c$$)

(ii)  P($$A\cap B^c$$) = P(A).P($$B^c$$)

(b)  Three events A,B and C are independent if & only if all the following conditions hold;

P($$A\cap B$$) = P(A).P(B);    P($$B\cap C$$) = P(B).P(C)

P($$C\cap A$$) = P(C).P(A) and P($$A\cap B\cap C$$) = P(A).P(B).P(C)

i.e. they must be pairwise as well as mutually independent.

(c)  If three events A, B and C are pair wise mutually exclusive i.e. P($$A\cap B$$) = P($$B\cap C$$) = P($$C\cap A$$) = 0 => P($$A\cap B\cap C$$) = 0. However the converse of this is not true.

Note :

Independent events are not generally mutually exclusive and vice versa. Mutually exclusiveness can be used when the events are taken from the same experiment and independence can be used when the events are taken from different experiments.

Example : If A and B are independent events such that P($$A\cap B^c$$) = $$1\over 3$$ & P($$A\cup B$$) = $$11\over 15$$,then P($$A\cap B$$) is equal to

Solution : P(A) – P($$A\cap B$$) = $$1\over 3$$

P($$A\cup B$$) = P(A) + P(B) – P($$A\cap B$$) = $$11\over 15$$

=> P(B) = $$6\over 15$$ = $$2\over 5$$

P(A) – P(A).P(B) = $$1\over 3$$ => P(A) = $$5\over 9$$

=> P($$A\cap B$$) = P(A).P(B) = $$2\over 5$$x$$5\over 9$$ = $$2\over 9$$

Hope you learnt dependent and independent events in probability and total probability theorem in probability. To learn more practice more questions and get ahead in competition. Good Luck!