Probability of two variables given in a table.
Conditional probability: $P(Y=y | X=x)$ means the probability that you get $y$ given that you know you got $x$.
$$ P(Y=y | X=x) := \frac{P(Y=y \text{ and } X=x)}{P(X=x)}. $$
Usually used like this:
$$ P(Y=y \text{ and } X=x) = P(X=x) \cdot P(Y=y | X=x) .$$
Example when $E[X\cdot Y] \not= E[X]\cdot E[Y]$.
Independence. The events $X$ and $Y$ are independent if knowing $X$ gives you no information about $Y$. That is, $$P(Y=y | X=x) = P(Y=y) .$$
If $X$ and $Y$ are independent, then $E[X \cdot Y] = E[X] \cdot E[Y]$. Know the critical step where independence is used. The rest of the argument is not as important.
Example when $\Cov(X,Y)$ is not zero using numbers in a table.