the radioactive decay law
Watch a lump of radioactive material and you cannot say when any particular nucleus will decay: it might go in the next second or in a billion years, and its age makes no difference to the odds. Yet out of a huge number of such nuclei, an utterly predictable fraction decays per unit time. The radioactive decay law is the simple mathematics that turns perfect individual randomness into perfect collective regularity.
The single assumption is that each nucleus has a constant probability per unit time of decaying, the decay constant lambda, independent of its history. For N undecayed nuclei this gives dN/dt = -lambda N, whose solution is exponential: N(t) = N_0 e^(-lambda t). Three quantities describe the same fact. The half-life t_1/2 = ln 2 / lambda is the time for half the sample to decay. The mean lifetime tau = 1/lambda is the average survival time. The activity A = lambda N (measured in becquerels, one decay per second) is the number of decays per unit time and itself decays exponentially.
This law underpins radiometric dating (carbon-14 for archaeology, uranium-lead for rocks), medical tracer dosing, and reactor and waste management. Its foundation is genuinely quantum: the decay is a spontaneous quantum transition (alpha decay by tunnelling, beta decay by the weak interaction), and the constant per-unit-time probability makes the process memoryless. The exponential is exact only in the ideal large-number limit; for a few atoms you see statistical fluctuations around it, and it is not literally the decay 'schedule' of any single nucleus.
Carbon-14 has t_1/2 = 5730 years, so lambda = ln 2 / 5730 per year, about 1.21e-4 per year. After 11460 years (two half-lives) a quarter of the original carbon-14 remains, the basis of radiocarbon dating.
Each half-life removes half of whatever is left, forever approaching but never reaching zero.
The law is statistical. An individual nucleus does not 'age' toward decay, and its probability of decaying in the next instant never changes; the smooth exponential emerges only from averaging over many nuclei.