A radioactive atom has no memory and no schedule — it might decay in the next second or in a billion years. Yet a CROWD of them is perfectly predictable: in one half-life, exactly half are gone; in another, half of what’s left; forever halving but never reaching zero. Slide the clock and watch the crowd thin.
Decay is a memoryless (Poisson) process: each nucleus has a fixed probability per unit time, so the surviving fraction is N(t)/N₀ = 2^(−t/T) = e^(−λt), where T is the half-life and λ = ln2/T. After one half-life exactly 1/2 remain, after two 1/4, after three 1/8 — exponential, never quite zero. The instrument tracks a decaying population. A fail-loud self-check throws unless the surviving fraction is 1/2 at one half-life, 1/4 at two, and strictly decreasing.
The smooth curve is the LARGE-crowd limit; a handful of atoms decays in random jumps around it (radioactivity is genuinely stochastic). The 2^(−t/T) law and the halving are exact for the ensemble.