Find Mistake: Independence of two Events

Question

Assume we have a black and a red cube with 6 sides. We definite two events A = "the black dice shows 5", B = "The product of the number of pips is a prime number". We roll the dice. So $P[A] = \frac{1}{6}$ and $P[B] = \frac{1}{6}$ right ? Now i want to check, if A and B are independent. So $P[A \cap B] = \frac{1}{36} = \frac{1}{6} * \frac{1}{6} = P[A]P[B]$ so A and B are independent. My instincts tell me the events are dependent. Where is my mistake ?

Answer 1

$B$, "The product of the number of pips is a prime number", is the event $\{(1,2),(1,3),(1,5),(2,1),(3,1),{\bf(5,1)}\}$

So $\mathsf P(A\mid B)$ is clearly $1/6$.

The weighted ratio of outcomes in $A\cap B$ to $B$ equals the weighted ratio of outcomes in $A$ to $\Omega$ (the outcome set).

That is all that is required for independence.

The notion that "independent events don't influence each other" is not too misleading.   It is just that our judgement on whether this is the case is not very reliable.   Our instincts can be way off.

Answer 2

Your mistake, probably, is that you're forgetting the fact that both the dice are rolled SIMULTANEOUSLY.

This problem is somewhat analogous to another problem where two dice are rolled simultaneously, and the number appearing on the first die is independent of the number appearing on the second one.