This document contains the common definitions that may be asked in the Cambridge IGCSE examinations.
Radioactive decay is both random over space and time, and radioactive decay is spontaneous.
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.