Seven half-lives take a radioactive sample below 1%, and ten leave about a thousandth.
Each half-life halves what is left, so the fraction is one half to the power of the number of half-lives: seven gives 0.78%, ten gives 0.098%, twenty about one in a million. It is why a sample is treated as gone after about ten half-lives, and why carbon dating, which needs a measurable amount of carbon-14, stops at about nine (51,600 years).
It is the law for large numbers: a handful of atoms may all be gone after two half-lives or last through five. It never reaches exactly zero, so a hazard with a large starting activity can stay a hazard past ten half-lives. It also assumes one decay step; where a nuclide decays into another radioactive one (radium into radon, for example), the daughter keeps the sample active.
Estimate with the rule, then check it against the calculator that models it properly.
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