[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.

[BobaFloutist]

Why is the most likely time right now? What makes right now more likely than in five minutes? I guess you're saying if there's nothing that makes it more likely to fail at any time than at any other time, right now is the only time that's not precluded by it failing at other times? I.E. it can't fail twice, and if it fails right now it can't fail at any other time, but even if it would have failed in five minutes it can still fail right now first?

[froobius]

Yes that's pretty much it. There will be a decaying probability curve, because given you could fail at any time, you are less likely to survive for N units of time than for just 1 unit of time, etc.

[tsimionescu]

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

This is a completely different argument that relies on various real-world assumptions, and has nothing to do with the Copernican principle, which is an abstract mathematical concept. And I actually think this does make sense, for many common categories of processes.

However, even this estimate is quite flawed, and many real-world processes that intuitively seem to follow it, don't. For example, looking at an individual animal, it sounds kinda right to say "if it survived this long, it means it's robust, so I should expect it will survive more". In reality, the lifetime of most animals is a binomial distribution - they either very young, because of glaring genetic defects or simply because they're small, fragile, and inexperienced ; or they die at some common age that is species dependent. For example, a humab that survived to 20 years of age has about the same chance of reaching 80 as one that survived to 60 years of age. And an alien who has no idea how long humans live and tries to apply this method may think "I met this human when they're 80 years old - so they'll probably live to be around 160".

[froobius]

Ah no, it is the Copernican principle, in mathematical form.