Physical predictions involving identical particles

We'll treat the problem only for a two-particle system. Consider two identical particles which can be in the individual states ,which include the spin-dependence and are orthogonal. The physical state of the system is:

Suppose we want to measure on each particle the physical quantity
What is the probability of finding for one particle and for the other? The final state is:
The desired probability is given by for bosons and for fermions, that is:
The term is called direct term, while the term is called exchange term:
This shows the appearance of a sort of interference phenomenon when we're dealing with identical particles.