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.
Basically, when we don’t see any members of the species for a while. They could be extinct, or they could be good at hiding. The Zanzibar leopard is an example that is or was thought to be extinct, but there is some evidence to suggest they may still be alive and just really good at living around humans without being seen.
I know that when you’re investigating how many different species there are in an enviroment, you can use [rarefaction](https://en.m.wikipedia.org/wiki/Rarefaction_(ecology)) in order to determine whether or not you have sampled enough of the enviroment to get an accurate look of that area’s richness. I’m not an expert, but I would imagine that a similar thing can be done with a specific species in order to tell how likely it is to be extinct
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.
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