We can’t be sure, that’s why some species “come back to life” after years of supposed extinction.
>*Evolutionary biology isn’t always a perfect science. Researchers themselves have even given a name to species that they mistakenly believed to be extinct. Named for the legendary Lazarus who came back from the dead, Lazarus species are the wide variety of animals that have seemed to do the same thing. They’ve gone from officially not existing to existing once again. It’d be miraculous if they hadn’t actually been alive all along. When these creatures had gone unseen for years upon years, scientists just wrote them off. When they reappeared (sometimes more than 100 years after last being seen!) they totally shocked the world. We can only hope that some of today’s endangered and extinct species will pleasantly surprise us with Lazarus status in the future.*
[Whole article](https://www.thetravel.com/14-creatures-scientists-thought-were-extinct-but-arent-11-gone-forever/)
Take the Northern White Rhino. It is big enough that we know exactly how many are left: two females, neither of whom could carry a calf to term, no living male. It’s extinct, with the caveat that scientists MIGHT be able to produce an embryo from frozen sperm and eggs that MIGHT implant in a southern white rhino, and MAYBE carry to term. But for all intents and purposes, it’s gone.
Littler creatures can be deemed to be gone if X number of years go by and none have been reported, and it is known to have a limited range. We know for example that orangutans have a very small range, constantly being lost to palm plantations. There will still be a population in captivity when the last wild ones are gone, but probably not enough to provide adequate genetic diversity.
As everyone else has said, we don’t. One of the poster child cases for captive breeding programs is the [black footed ferret](https://en.m.wikipedia.org/wiki/Black-footed_ferret) which was thought to be extinct until a population of 18 was discovered. Now the population is over 1000 in multiple locations to help protect them from complete extinction.
More recently, [the widely thought extinct Vietnamese mouse deer](https://www.cnn.com/2019/11/11/asia/mouse-deer-vietnam-chevrotain-rediscovered-scn/index.html) was captured on game cameras by biologists after having not seen it for 30 years.
Capture/recapture experiments can help. Say you capture 10 rhinos and spraypaint a dot on them. Release them, come back a year later and capture 10 rhinos again. Only 1 of them has spraypaint, which means that you capture 10% of the population, and the population is 100.
You can use that tool and some other fancy predictive modeling to track a population over time and be able to accurately estimate the population even when it gets very small.
There is never any way to know for sure, though.
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