Come up with an estimate for how many stars are likely to have habitable planets. Then of those, how many will form life. Then of those, in how many will that life become multicellular, then complex, then sentient, then sapient, then civilized, then industrial, then nuclear, then space-faring, and so on.
Take the combined percentages of all those things, multiply by an estimated number of stars in the entire universe, and according to those estimates, you should come up with something like hundreds of thousands of galactic-level civilizations popping up over time leaving evidence everywhere that alien life exists.
But we have no evidence of any of them existing. That’s the paradox.
edit – and sorry, this was a fairly quick reply that melded the answer with the Drake Equation and I could have gone into a lot better detail. Wasn’t expecting it to blow up
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