time left = time so far
But as you note prior knowledge will enable a better guess.
> I visited the Berlin Wall. People at the time wondered how long the Wall might last. Was it a temporary aberration, or a permanent fixture of modern Europe? Standing at the Wall in 1969, I made the following argument, using the Copernican principle. I said, Well, there’s nothing special about the timing of my visit. I’m just travelling—you know, Europe on five dollars a day—and I’m observing the Wall because it happens to be here. My visit is random in time. So if I divide the Wall’s total history, from the beginning to the end, into four quarters, and I’m located randomly somewhere in there, there’s a fifty-percent chance that I’m in the middle two quarters—that means, not in the first quarter and not in the fourth quarter.
> Let’s suppose that I’m at the beginning of that middle fifty percent. In that case, one-quarter of the Wall’s ultimate history has passed, and there are three-quarters left in the future. In that case, the future’s three times as long as the past. On the other hand, if I’m at the other end, then three-quarters have happened already, and there’s one-quarter left in the future. In that case, the future is one-third as long as the past.
https://www.newyorker.com/magazine/1999/07/12/how-to-predict...
What this "time-wise Copernican principle" gives you is a guarantee that, if you apply this logic every time you have no other knowledge and have to guess, you will get the least mean error over all of your guesses. For some events, you'll guess that they'll end in 5 minutes, and they actually end 50 years later. For others, you'll guess they'll take another 50 years and they actually end 5 minutes later. Add these two up, and overall you get 0 - you won't have either a bias to overestimating, nor to underestimating.
But this doesn't actually give you any insight into how long the event will actually last. For a single event, with no other knowledge, the probability that it will after 1 minute is equal to the probability that it will end after the same duration that it lasted so far, and it is equal to the probability that it will end after a billion years. There is nothing at all that you can say about the probability of an event ending from pure mathematics like this - you need event-specific knowledge to draw any conclusions.
So while this Copernican principle sounds very deep and insightful, it is actually just a pretty trite mathematical observation.
Edit: I should add that, more specifically, this is a property of the uniform distribution, it applies to any event for which EndsAfter(t) is uniformly distributed over all t > 0.
The cumulative distribution actually ends up pretty exponential which (I think) means that if you estimate the amount of time left in the outage as the mean of all outages that are longer than the current outage, you end up with a flat value that's around 8 hours, if I've done my maths right.
Not a statistician so I'm sure I've committed some statistical crimes there!
Unfortunately I can't find an easy way to upload images of the charts I've made right now, but you can tinker with my data:
cause,outage_start,outage_duration,incident_duration
Cell management system bug,2024-07-30T21:45:00.000000+0000,0.2861111111111111,1.4951388888888888
Latent software defect,2023-06-13T18:49:00.000000+0000,0.08055555555555555,0.15833333333333333
Automated scaling activity,2021-12-07T15:30:00.000000+0000,0.2861111111111111,0.3736111111111111
Network device operating system bug,2021-09-01T22:30:00.000000+0000,0.2583333333333333,0.2583333333333333
Thread count exceeded limit,2020-11-25T13:15:00.000000+0000,0.7138888888888889,0.7194444444444444
Datacenter cooling system failure,2019-08-23T03:36:00.000000+0000,0.24583333333333332,0.24583333333333332
Configuration error removed setting,2018-11-21T23:19:00.000000+0000,0.058333333333333334,0.058333333333333334
Command input error,2017-02-28T17:37:00.000000+0000,0.17847222222222223,0.17847222222222223
Utility power failure,2016-06-05T05:25:00.000000+0000,0.3993055555555555,0.3993055555555555
Network disruption triggering bug,2015-09-20T09:19:00.000000+0000,0.20208333333333334,0.20208333333333334
Transformer failure,2014-08-07T17:41:00.000000+0000,0.13055555555555556,3.4055555555555554
Power loss to servers,2014-06-14T04:16:00.000000+0000,0.08333333333333333,0.17638888888888887
Utility power loss,2013-12-18T06:05:00.000000+0000,0.07013888888888889,0.11388888888888889
Maintenance process error,2012-12-24T20:24:00.000000+0000,0.8270833333333333,0.9868055555555555
Memory leak in agent,2012-10-22T17:00:00.000000+0000,0.26041666666666663,0.4930555555555555
Electrical storm causing failures,2012-06-30T02:24:00.000000+0000,0.20902777777777776,0.25416666666666665
Network configuration change error,2011-04-21T07:47:00.000000+0000,1.4881944444444444,3.592361111111111