When arriving at a site of a surface, an adatom has three options. There is a probability that it will adsorb to the surface (), a probability that it will migrate to another site on the surface (), and a probability that it will desorb from the surface and return to the bulk gas (). For an empty site (&theta;=0) the sum of these three options is unity.
For a site already occupied by an adatom (&theta;>0), there is no probability of adsorbing, and so the probabilities sum as:
For the first site visited, the P of migrating overall is the P of migrating if the site is filled plus the P of migrating if the site is empty. The same is true for the P of desorption. The P of adsorption, however, does not exist for an already filled site.
The P of migrating from the second site is the P of migrating from the first site and then migrating from the second site, and so we multiply the two values.
Thus the sticking probability () is the P of sticking of the first site, plus the P of migrating from the first site and then sticking to the second site, plus the P of migrating from the second site and then sticking at the third site etc.
There is an identity we can make use of.
The sticking coefficient when the coverage is zero  can be obtained by simply setting . We also remember that  
If we just look at the P of migration at the first site, we see that it is certainty minus all other possibilities.
Using this result, and rearranging, we find: