Zero-One Laws: Examples of Measurable Events w/ Probability 0 or 1

[economicsnerd]

There are various zero-one laws (e.g. Kolmogorov's) that assure us that certain measurable events have probability zero or one in a given context.

Does anybody know any good examples of events (preferably "naturally" occurring elsewhere in math) which are covered by such a theorem, but for which it's currently an open question whether the probability is zero or one?

 

[eigenperson]

http://mathoverflow.net/questions/2...e-law-gives-probability-0-or-1-but-hard-to-de

A simple example: consider the 3-dimensional cubic grid, and connect each point to the 6 adjacent ones by edges. Then delete each edge with probability p.

Define a "cluster" to be a connected component of the resulting graph.

Let f(p) be the probability that there exists an infinite cluster. By the zero-one law, f(p) is either 0 or 1, since the existence of an infinite cluster does not depend on the edges in any finite box. Also, it should be clear that increasing p cannot increase the probability of an infinite cluster, so that f is nonincreasing. Therefore, the only question is at which critical probability f switches from 1 to 0. The exact value is unknown (though there are good estimates).

 

[economicsnerd]

Any others?

 

[jk22]

In physics there are hidden variables controversies you can prove they cannot be probabilistic if there exist a single case were the correlation is exact : https://www.physicsforums.com/threads/measurement-problem-and-computer-like-functions.828840/page-3