To show that events A and B are independent, we need to demonstrate that the occurrence of one does not affect the occurrence of the other. By definition, two events A and B are considered independent if:
- P(A and B) = P(A) * P(B)
Now, let’s understand why the independence of A and B implies the independence of A and B:
Since A and B are independent, we know:
- P(A and B) = P(A) * P(B)
Additionally, we can express the probability of the intersection of A and B in terms of P(A | B) (the probability of A given B) and P(B):
- P(A and B) = P(A | B) * P(B)
Because A and B are independent, we also have:
- P(A | B) = P(A)
Substituting this into our equation gives us:
- P(A and B) = P(A) * P(B)
This confirms that A and B, being independent events, indeed means that A and B are also independent. Therefore, we validate the statement: if A and B are independent events, then A and B are also independent.