Suppose that a fair dice is rolled and that the number $x$ appears. Let $E_1$ be the event that the number $x$ is even, $E_2$ be the event that the number $x$ is greater than or equal to $3$, $E_3$ be the event that the number $x$ is a $4,5$ or $6$. Are the events $E_1$ and $E_2$ independent? Are the events $E_1$ and $E_3$ independent?
I would say events $E_1$ and $E_2$ are independent because
$\mathbb{P}(E1 \cap E2) = \mathbb{P}(E1) \cdot \mathbb{P}(E2) \Leftrightarrow \mathbb{P}(\{4,6\}) = \mathbb{P}(E1) \cdot \mathbb{P}(E2) \Leftrightarrow \frac{1}{3} = \frac{1}{2} \cdot \frac{4}{6}$
You initially got it backwards. Because, for example,
$$\mathbb{P}(E1 \cap E2) \neq \mathbb{P}(E1) \cdot \mathbb{P}(E2),$$
they're dependent, or not independent.
But, if the joint probability equals the product of the individual probabilities, then the events are independent.