How can I show that two events are independent ?

Two events are independent if and only if the product of their own probabilities is equal to the probability of them both happening, that is :
A and B independent <=> P(AnB) = P(A)*P(B)
NB : This works also for a larger number of events

Let's consider two examples :
1) p(A) = 0.5, p(B)=0.3, p(AnB)=0.15In that case,p(A)p(B) = 0.50.3 =0.15, which is equal to p(AnB),so A and B are independent
2) p(A) = 0.2, p(B)=0.4, p(C)=0.1, p(AnBnC)=0.05In that case,p(A)p(B)p(C) = 0.20.40.1 = 0.008, which isn't equal to p(AnBnC),so A and B are not independent