Tried looking it up and it got complicated real quick.
In: Physics
You have a chunk of U238 and half of it will have decayed in 4.5 billion years. So 1/4 decays in 2.25 billion years 1/8 in 1 billion years 1/16 in 500 million years 1/32 in 250 million years 1/64 in 125 million… 1/(2^(50)) = ~ 1/(1.26×10^(15)) in 4×10^(-6) years = ~ 2 minutes. The chunk of U238 contains around 10^(23) atoms so 1/(2^(50)) of that is ~10^(7) number of atoms.
Since atoms are plenty in the sample with a half life of 4.5 billion years you still get (with this naive estimation) 10 million or so decays under about 2 minutes. So you can measure for some trivial amount of time and get the decay rate of the matterial. Thats how many decays per unit time tend to happen. Even though the decay rate tells you how many decays per unit time happen on average with things like this the law of large numbers work well so the fluctuations around the measured averages is negligible. Half life is what you get when either you take N atoms and ask how long do you have to wait on average to get N/2 or because N isn’t really a factor here you can say how long do you have to wait for one atom to have a 50% chance of being not decayed (or decayed its 50-50). Its just when you have N many atoms the probabilities turn to frequencies.
So with N many atoms 4.5 billion years of half life isn’t that long but sometimes matterials can have insanely long half life and in that case you need to run an experiment for a year or two to observe a dozen or so decays in a reasonable sized sample.
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