thesaurus.maths.org

Two variables are independent if one can be varied without affecting the other.
Two events are independent if the probability distribution of one is not affected by the result of the other one.
E.g. When you throw a dice, the score you get on your second throw is not affected by whatever score you got on the first throw.

Two events A and B are independent if

P(A ∩B) = P(A) P(B);

a collection of events { A_i | i ∈ I } is independent if

P   ∩ i ∈ J  Ai   = ∏ i ∈ J  P(Ai)

for all finite subsets J of I .
Discrete random variables X₁, …,X_n are independent if, for any x₁, …,x_n ,

P(X1=x1, …,Xn=xn) = n ∏ i=1  P(Xi = xi).