[froobius]

> while this Copernican principle sounds very deep and insightful, it is actually just a pretty trite mathematical observation

It's important to flag that the principle is not trite, and it is useful.

There's been a misunderstanding of the distribution after the measurement of "time taken so far", (illuminated in the other thread), which has lead to this incorrect conclusion.

To bring the core clarification from the other thread here:

The distribution is uniform before you get the measurement of time taken already. But once you get that measurement, it's no longer uniform. There's a decaying curve whose shape is defined by the time taken so far. Such that the estimate `time_left=time_so_far` is useful.

[tsimionescu]

If this were actually correct, than any event ending would be a freak accident: since, according to you, the probability of something continuing increases drastically with its age. That is, according to your logic, the probability of the wall of Berlin falling within the year was at its lowest point in 1989, when it actually fell. In 1949, when it was a few months old, the probability that it would last for at least 40 years was minuscule, and that probability kept increasing rapidly until the day the wall was collapsed.

[froobius]

That's a paradox that comes from getting ideas mixed up.

The most likely time to fail is always "right now", i.e. this is the part of the curve with the greatest height.

However, the average expected future lifetime increases as a thing ages, because survival is evidence of robustness.

Both of these statements are true and are derived from:

P(survival) = t_obs / (t_obs + t_more)

There is no contradiction.