What you are looking at
A sample of radioactive nuclei (green). Each one carries the same fixed probability of decaying per second —
a genuine quantum coin-flip with no memory and no warning. When one decays it flashes gold and turns into a
stable daughter (violet). The graph races the jagged, measured population against the smooth exponential
prediction.
From randomness to law
If each nucleus decays with probability λ per unit time, a population N loses λN atoms per second on
average:
dN/dt = −λN ⇒ N(t) = N₀·e^(−λt)
No individual event is predictable — but the average over hundreds of atoms is locked in. The wiggles you
see around the gold curve are real statistical noise, of size about √N; watch them matter more and more as
the sample shrinks.
The half-life
The time for
half of whatever remains to decay is the same at every moment — decay has no memory:
T½ = ln 2 / λ ≈ 0.693 / λ
After 1 half-life 50% remain; after 2, 25%; after n, (½)ⁿ. The dashed steps on the graph mark exactly this
halving staircase. Note what a half-life is
not: the sample never dies "on schedule" — it just
keeps halving, which is why tiny traces of long-lived isotopes linger essentially forever.
Activity: what a Geiger counter hears
A detector doesn't count atoms — it counts decays per second, the
activity:
A = λN = A₀·e^(−λt)
Activity decays with the same half-life as the population, which is what makes radiometric dating work:
measure today's activity of carbon-14 (T½ = 5,730 yr) in a bone, compare with a living sample, count the
halvings — and you've counted the millennia.
Things to try
Run a small sample (N₀ = 50) a few times and notice each run takes a visibly different path — then run 1000
and watch the noise almost vanish: √N grows slower than N. Pause at exactly one half-life and check ~50%
survive. Let it run to the last few atoms and ask when the final one will decay: the honest answer is that
nobody — not even the atom — knows.