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.

If you are given a single radioactive nuclei, it would be impossible to know whether this particular nuclei would decay after one half-time. It might decay immediately or after more than one half-life. It is important to understand that you can observe approximately half of all the original nuclei decay after one half-life only if you have a large sample of nuclei.

### Spontaneous

Spontaneous means that a radioactive decay is unaffected by external conditions. You cannot change the rate of decay by applying more pressure or higher temperature. You also cannot change the rate of decay by including more radioactive nuclei. The probability of decay is a constant which is determined by the type of nuclei.

### Simulation

This link shows a simulation to demonstrate the randomness of radioactive decay. Run two simulations and observe that after 1.00T(one half-life), the number of decayed nuclei would not be the same between the two simulation runs, although the numbers are close. This demonstrate that radioactive decay is random.