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.
Note that, if A,B are independent events then P(B/A) = P(B) and P(A/B) = P(A)
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:
- Two events A,B are independent.
- Two events A, B are independent.
- Two events A, B are independent.
- 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.