How are billion-year half-lives approximated?

Question

Our life-time is negligible, compared to that of unstable nuclei with half-lives of billions of years.

How, then, are these half-lives estimated, if you may never even witness a decay. It's like estimating the probability of a popcorn kernel popping, after one of the 200 in a pan has popped.

Would it be done over, say, a decade, with a massive amount of the source constantly monitored by a gygameter? Wouldn't the data still be inconclusive?

Is it estimated other ways, such as by examining the relationship between the composition of an atom and its half-life, to extrapolate larger half-lives? Comparing the quantities of the nucleuses isotopes found in nature, to extrapolate?

-- How on earth could scientists know that Lead-204 has a half-life 10,000,000 time longer than the age of the universe ~14,000,000,000 so: 140,000,000,000,000,000 years.

Answer 1

Because there are a lot of atoms. Lets take the example of lead 204. If we take 1 Mol, which is 204g we have $6×10^{23}$ atoms. You say half-life is $140×10^{15}$years which is around $4×10^{24}$ seconds. So in a crude estimation you still should have around 1 decay every ten seconds. In order to find this, you look for the outgoing radiation, not for the decay product.