Independent Events of Probability

Independent Events

Two events A, B are said to be independent if and only if the Probability of (A intersection B) equals Probability of A multiplied to Probability of B.

Three elements A,B,C are independent if and only if
P(A∩B∩C) = P(A). P(B). P(C)

The following statements are equivalent which means everyone of them implies the remaining three:

1. Two events A,B are independent.
2. Two events A, B are independent.
3. Two events A, B are independent.
4. Two events A, B are independent.

2 → 3
Suppose A, B are independent. This means P(A ∩ B) = P(A). P(B)

Consider P(A ∩ B) = P (B – A)

= P(B) – P(A ∩ B)

= P(B) – P(A) . P(B)

= P(B) (1 – P(A))

= P(A).P(B)

Therefore A, B are independent.

3 → 4
Suppose A, B are independent.
so, P(A∩ B) = P(A).P(B)

P(A ∩ B) = P(A ∪ B )

= 1 – P(A ∪ B)

= 1 – P(A) – P(B) + P(A ∩ B)

= P(A) – P(B) + P(A) – P(B)

= P(A) – P(B) (1 – P(A))

= P(A) (1 – P(B))

= P(A). P(B)

Therefore A, B are independent.

4 → 1
Suppose A, B are independent.
so we have P(A ∩ B) = P(A) . P(B)

P(B – (A∩B)) = (1 – P(A)) . P(B)

P(B) – P(A∩B) = P(B) – P(A) . P(B)

P(A∩B) = P(A) . P(B)

Therefore A,B are independent.

Therefore we can conclude that the above four statements are equivalent.