John has again a red and blue dice and throws them once. For i = 1, 2, 3, let the events Ai be A1: the outcome of the red die is 1, 2, or 5;

A2: the outcome of the red die is 4, 5, or 6;
A3: the sum of the two outcomes is equal to 9.
Prove that P(A1A2A3) = P(A1)P(A2)P(A3)
while each of the pairs {A1, A2}, {A1, A3}, and {A2, A3} consist of dependent events.