N people are seated around a (round) table. What's the probability that fixed two people sit next to each other?

Let $\Omega$ be a set of possible outcomes of an experiment. Let $\mathcal{F}$ be a collection of subsets of $\Omega$ that are closed under countable unions. What could be properties that allow us to define a probability function $P \colon \mathcal{F} \to [0,1]$? If we remove the constraint that $\mathcal{F}$ is not closed under countable unions, can we define a probability function?

In an experiment involving two events, $A$ and $B$ are known to have $P(A) = 0.5$ and $P(B) = 0.65$. It is suspected that these two events can be independent. However $P(A \cap B)$ can not be determined by means of theory. So you run an experiment $500$ times and record that $A \cup B$ occurs 497 times. What would you conclude regarding independence between $A$ and $B$?