Since naturally occurring uranium (U-238) has a half life of 4.5 billion years, then it means half of the uranium on earth has decayed into lead by now. But why only half, and why that specific half? What was special about the particles that did decay? Were they different in some way?
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Radioactive particles constantly play the lottery of quantum mechanics (or physics in general). When they hit their jackpot, they decay. This means two things: decays happen randomly, and decays happen independent of other particles.
The result is an exponential decay, which can be formulated into a halftime. You can also reformulate the exponential decay into a thirdtime or decitime or whatever else is convenient.
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