Question

For a spontaneous process in an isolated system, the change in entropy is positive. Why? Why not?

Answer 1

Hint: In physical science, an isolated system is either: a physical system that is so far separated from other systems that it does not interact with them; or a physical system that is so far removed from other systems that it does not interact with them.
a thermodynamic system surrounded by immovable hard barriers through which no mass nor energy may pass
An isolated system is typically assumed to be outside the reach of external gravitational and other long-range forces, yet being subject to its own gravity.

Complete answer:
A spontaneous process in thermodynamics is one that happens without any external input to the system. The time-evolution of a system in which it releases free energy and goes to a lower, more thermodynamically stable energy state is a more technical definition (closer to thermodynamic equilibrium). The sign convention for free energy change is the same as for thermodynamic measurements, with a release of free energy from the system resulting in a negative change in the system's free energy and a positive change in the free energy of the surroundings.
Because the second principle of thermodynamics states that in an isolated system, the change in entropy can either be positive or equal zero, the process is reversible. However, the process must be in equilibrium for every little (infinity small in theory) change in the process to be reversible.
Because your process is spontaneous, there is no equilibrium between the initial state and the final state in your situation.
Spontaneous processes are characterised by a rise in entropy when they involve an isolated system with no energy exchange with the environment. Under the conditions of interest, a spontaneous reaction is a chemical reaction that occurs spontaneously.

Note:
In general, a process's spontaneity simply affects whether or not it can happen, not whether or not it will happen. To put it another way, spontaneity is a required but not sufficient condition for a process to take place. Furthermore, spontaneity has little bearing on the rate at which a spontaneous event might occur. The spontaneous conversion of a diamond into graphite, for example, occurs at ambient temperature and pressure. This process does not occur, despite being spontaneous, since the energy required to break the strong carbon-carbon bonds is more than the free energy released.