TITLE: Intuition of Kolmogorov‘s 0-1 Law
QUESTION [7 upvotes]: Kolmogorov‘s 0-1 Law seems to say something very intuitive: Tail events of independent objects have probability $0$ or $1$.
At least it should be intuitive - but it is not to me. Why exactly are tail events supposed to have this behaviour? What‘s the intuitive argument?

REPLY [2 votes]: Intuition is right there in the proof of the theorem. If $A \in \sigma (X_n,X_{n+1},...)$ for every $n$ then $A$ is independent of $X_1,X_2,..,X_n$ for every $n$. So what does $A$ depend on? Intuitively it is independent of itself and only trivial events can be independent of themselves.