Could every atom in a sample with a five-year half-life happen to decay in the next minute?

There is a deeper point hiding in your discomfort, and it is worth naming: the model is a model.

Exponential decay with a constant probability per unit time is an extremely well-tested description, and it is a description. At the extreme tails, where you are asking about events with probabilities like the one above, we have no experimental access whatsoever and never will. Nobody has ever observed the tail of that distribution and nobody could.

So strictly, the answer "yes, with vanishing probability" is a statement about the model, not a verified statement about the world. Whether the model holds to twenty decimal places in the tail is unknowable by any experiment we can perform.

That is not a reason to distrust it. It is a reason to be precise about what is being claimed, and your instinct that something odd is going on is picking up on exactly that.

The practical version of this that does matter: the statistics stop being reliable when the number of atoms gets small.

With a kilogram of material, the observed decay rate matches the predicted curve to extraordinary precision, because averaging over that many independent events leaves essentially no fluctuation. With a few hundred atoms, the fluctuations are visible and the smooth exponential curve is only an average.

With a single atom, half-life means only that there is a fifty percent chance it has decayed by then. It may sit there for a thousand half-lives. That case is not exotic — it is the everyday situation in single-atom experiments, and it is where the statistical nature of the law stops being an abstraction.

Your reasoning is correct. It is possible, and the probability is small in a way that is genuinely difficult to convey.

Each nucleus decays independently, with a fixed probability per unit time. For a five-year half-life, the chance that a given nucleus decays in any particular minute is roughly one in two million. The chance that all of them do is that number multiplied by itself once for every nucleus present.

A kilogram of material contains something on the order of a billion billion billion nuclei. So the probability is a number like one in two million, raised to a power with twenty-something digits in the exponent.

Written out, that probability has more zeros after the decimal point than there are atoms in the observable universe. It is not merely astronomically small; it is small on a scale where the word astronomical is far too weak.

Worth adding the flip side, since people often assume it: the probability per unit time genuinely is constant. A nucleus does not age.

A nucleus that has existed for a billion years is exactly as likely to decay in the next minute as one created a second ago. There is no internal clock winding down, no accumulated wear. That property is what makes the exponential the right functional form, and it is one of the more counter-intuitive facts in physics.