# What's the intuition behind Kolmogorov zero-one law? Why can't the probability be a number in between?

I'm trying to understand intuitively why the probability of a tail event must be zero or one. Why can't it be some real number? I'm sure that when Kolmogorov proved this theorem he must have postulated it using intuition first.

• Could you please give us a link to that "Kolmogorov 0-1 law"? What does "tail event" mean? Commented May 23, 2020 at 17:04
• You have $\mathcal F_\infty =\bigcap \mathcal F_{\ge n}$. To know whether or not $A \in \mathcal F_{\ge n}$ it is sufficient to have the knowledge of future starting from some time $n$. Since information about "certain point" is independent of the others, then by knowing the future $\ge n$ you don't have any knowledge about past. Now, $A \in \mathcal F_\infty$ if that $n$ is arbitrary large, so in $A$ you shouldn't have any interesting information. Commented May 23, 2020 at 17:25
• @user247327: en.wikipedia.org/wiki/Kolmogorov%27s_zero%E2%80%93one_law. It should be discussed in every beginning textbook on measure-theoretic probability. Commented May 23, 2020 at 17:28
• My intuition for this theorem is "in the long run, nothing is random anymore". Commented May 23, 2020 at 17:29

I don't know what Kolmogorov was thinking, but the theorem is an intuitive consequence of the idea that an infinite-dimensional random sequence $$(X_i)_{i\geq0}$$ is determined by the behavior of finite-dimensional subsets of elements $$(X_{i_1},\dots,X_{i_n})$$. That is, there's no "hidden behavior" of the sequence that you can't observe by looking at finite-dimensional subsets of the sequence.
So, if all the behavior depends on finite-dimensional subsets, then a "tail event" which, by definition, can't directly depend on any finite-dimensional subset, can only measure something "interesting" about the sequence through an indirect dependence on earlier finite subsets of elements. This can occur through inter-element dependence. For example, if every 100th element is the same, a "tail event" can depend indirectly on a finite subset of elements (specifically the single element $$X_{100}$$) through the dependence of infinitely many later elements on that earlier element.