Rami Arieli: "The Laser Adventure" Chapter 2.7 page 1
2.7 Stimulated Emission

Atoms stay in an excited level only for a short time (about 10-8 [sec]), and then they return to a lower energy level by spontaneous emission.

Every energy level has a characteristic average lifetime, which is the time after which only 1/e (about 37%) of the excited atoms still remain in the excited state. Thus, this is the time in which 67% of the excited atoms returned to a lower energy level.

According to the quantum theory, the transition from one energy level to another is described by statistical probability.
The probability of transition from higher energy level to a lower one is inversely proportional to the lifetime of the higher energy level.
In reality, the probability for different transitions is a characteristic of each transition, according to selection rules (For an explanation about selection rules click here).

When the transition probability is low for a specific transition, the lifetime of this energy level is longer (about 10-3 [sec]), and this level becomes a "meta-stable" level. In this meta-stable level a large population of atoms can assembled. As we shall see, this level can be a candidate for lasing process.

When the population number of a higher energy level is bigger than the population number of a lower energy level, a condition of "population inversion" is established.

If a population inversion exists between two energy levels, the probability is high that an incoming photon will stimulate an excited atom to return to a lower state, while emitting another photon of light. The probability for this process depend on the match between the energy of the incoming photon and the energy difference between these two levels (comment...)