The randomness of radioactivity and the wave/particle duality could simply not be fitted into the framework of classical physics. Yet it was clear that the classical picture worked very well for many purposes. The answer was soon provided by two brilliant theorists, Werner Heisenberg and Erwin Schrödinger who created the new science of quantum mechanics. Schrödinger constructed an equation and defined a wave function that led to the solution of a vast range of atomic problems. There were strong similarities to acoustics. We know that, in the open air, sound of any wavelength or frequency can be transmitted, but in an enclosed space, such as the interior of a room or the body of a wind instrument, only certain wavelengths and frequencies are possible. Similarly, in empty space electrons of any wavelength are possible, but the interior of an atom is like an enclosure, with rather soft walls defined by the attraction of the positive nucleus for the electrons. Electrons of less than certain energy cannot escape, and such electrons are restricted to certain discrete energies.
The wave and particle aspects of photons are complementary. Photons are detected as particles, at a particular point, but their motion from source to detector is described by a wave equation. Max Born proposed that Schrödinger's waves are waves of probability. Phenomena such as radioactivity and the double slit interference experiment show that individual events on the atomic scale can have a random property. Quantum phenomena have forced us to recognize that all kinds of individual events are not subject to strict causal laws, but the statistical behavior of large populations of identical atomic systems is rigorously predictable.