Conceptually, in the sense that nothing intrinsic to physics would prevent it, we could send a probe into a black hole.
Unfortunately the nearest black hole is about 1560 light years away from Earth. The fastest probe leaving our solar system ever launched (Voyager 1) is only going about 38,210 mph. That means it would take approximately 27.4 million years for a probe at that speed to reach said black hole. Not only would it not be working at that point but there almost certainly wouldn’t be anyone waiting to watch either. They wouldn’t be humans anyway so I guess “We” couldn’t watch even if we did manage to get it to go in eventually.
There also wouldn’t be much point to doing so. We can already see what happens when matter falls into black holes and that is about all we would get out of a probe, working or not. Once the probe passes beyond the event horizon we wouldn’t be able to get any signal from the probe even in theory, and conditions near the black hole would be extreme enough to rip anything apart. It isn’t a matter of “build it strong”, it is so extreme that things like protons and electrons would be stretched into spaghetti.
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