How to Determine Independence in a two die roll scenario?

Question

Assume event 1: we roll a 4 first, event 2: we roll a sum of 10.

The P(Event 1) = 1/6.

The P(Event 2) = 3/36 or 1/12. (3 ways out of 36 possibilities for 2 rolls to sum to 10).

To determine if events are independent, we need to check $P(A\cap B) = P(A) \cdot P(B)$.

So $P(Event 1 \cap Event 2) = 1/36 $ (only one way to roll a 4 first and get a sum of 10 out of 36 possibilities). Then $P(Event 1) \cdot P(Event 2) \neq 1/12$, so these events are dependent.

Is this logic correct or did I miss something?

Answer 1

I think you may want to end with

Then $P(Event 1) \cdot P(Event 2) = \frac16 \cdot \frac1{12} = \frac{1}{72}\not= \frac1{36} = P(Event 1 \cap Event 2)$ so these events are dependent.