Exponential decay curves appear in all scientific disciplines and are a way of studying/modeling samples changing with time in a simple manner.
A quantity undergoing exponential decay. Larger decay constants make the quantity vanish much more rapidly. This plot shows decay for decay constants (λ) of 25, 5, 1, 1/5, and 1/25 for x from 0 to 5.
This quantity may be used in a number of classical seteups as long as the supposition of :
the proportionality of the change in numbers with time as above , holds.
To get it experimentally many instances have to be recorded . In the case of particle decays it is a probability curve that appears as the solution of quantum mechanical equations. In the case of pharmacology, for example, it is a probability curve for the effects of a specific medication and the exponetial comes from simple assumptions of behavior. ( no esoteric differential equations).
In both cases it is useful in describing a sample, but useless in predicting a specific instance except statistically.
Let's suppose I'm in the lab and I claim that I can predict more than QM can, specifically, I can predict exactly at which moment in time a particle decays.
You have to find an individual particle, and wait for it to decay.
You don't believe me (naturally) so I set up the experiment, provide a piece of paper with a time written on it, and start the clock. At the time I have written down, the particle decays.
Exactly which of the six postulates of QM would this violate? As far as I can tell, it violates none of them so long as the results from multiple identical trials of this experiment reproduce the correct particle decay time distribution.
Then you need a second particle, a third, a fourth ..... Getting the exponential distribution is not the problem. The violation of the concept of Quantum Mechanics is in
At the time I have written down, the particle decays.
Either you are a metaphysical ( outside the experimental setup) seer or you have an alternative theory to Quantum Mechanics as QM only predicts probability curves, the Born rule.
Basically it is against the basic tenet that associates a measurement to the solutions of the Schrodinger equation via the square of the amplitude, as a probabilistic measurement.
As Floris said, throwing dice and claiming to know every throw goes against statistical probabilities. Your knowing every throw of the decay goes against quantum mechanical probabilities.
At the moment there are no successful theories to replace quantum mechanics, and the ones that have tried just reproduce it in a more convoluted, special and far more complicated ( as Bohm's model ) way.