[zamadatix]

For anyone confused by this response I had edited my comment after reading https://news.ycombinator.com/item?id=40740557 but before equivalence had hit reply and now their reply is left hanging. Sorry esquivalience! To summarize the linked answer on trusting the second dot isn't just randomly assigned: keep the context as physical post-its. Barring something like a matter bending psychic you'd be able to tell the dot under the second post-it was swapped as you made your pick.

That still leaves how to rely on chance of picks for a proof though.

[romwell]

It's the same thing as limits in spirit.

It's not that the chances of lying are small, it's that they can be made arbitrarily small.

Let's say my standards of "proof" are that there's only 0.1% chance that you're cheating. We play that game several times, and I'm satisfied.

Next comes someone else whose standard is 0.001% chance of cheating. They simply play the game a few more times, and they're satisfied too.

If they change their mind and decide that only 0.0000001% will make them happy, they simply tack on a few more rounds.

The key here is that the probability that you can cheat for arbitrarily long is exactly zero — for the same reason that Zeno's paradox is resolvable (and limit of 1, 1/2, 1/4, 1/8, 1/16, ... is exactly zero, and not just a very small number).

[zamadatix]

Great description in that "proof" in this context is more referring to the limiting behavior and being able to get to your desired level of arbitrary happiness than necessarily providing a traditional "proof" about it being a certainty within a finite amount of estimation. Thanks.

[romwell]

Thanks for the feedback, glad my comment was helpful!