Radioactive decay is both random over space and time, and radioactive decay is spontaneous.

### Random

Random over space means that given a large number of radioactive nuclei, it would be impossible to predict which nuclei would decay next. Random over time means that you cannot predict when a nuclei would decay at any time.

Given a large number of nuclei, it can be safe to predict that approximately half of the original radioactive nuclei would decay after one half-life. However, the exact number of nuclei decay would not be identical given two samples of equal number of original radioactive nuclei.