Although Richard's answer is accepted and correct, since he references the attitude in Kostka/Payne's Tonal Harmony, here's what another popular textbook has to say regarding the relative rarity of iii. From Harmony, fifth edition by Piston & DeVoto, Chapter 3 "Harmonic Progression in the Major Mode: Principles of Voice Leading":
The following generalizations are based on observations of usage by
composers in common practice. They are not proposed as a set of strict
rules to be rigidly adhered to.
(Emphasis mine, to show that this is intended as descriptive, rather than prescriptive.)
I is followed by IV or V, sometimes vi, less often ii or iii.
ii is followed by V, sometimes IV or vi, less often I or iii.
iii is followed by vi, sometimes IV, less often I, ii, or V.
IV is followed by V, sometimes I or ii, less often iii or vi.
V is followed by I, sometimes IV or vi, less often ii or iii.
vi is followed by ii or V, sometimes iii or IV, less often I.
vii° is followed by I or iii, sometimes vi, less often ii, IV, or V.
(Note that Piston & DeVoto do not follow the common convention of lower-case roman numerals for the minor triads; I've adjusted the above quote to avoid confusion about chord quality.)
In 4 of the 7 cases, the iii is in the "less often" column. In one case, it's in the "sometimes" column. Only after the vii° is it among the common choices
This list seems to reinforce the notion that the iii is rare, but not because it's to be avoided, only because it's special.
If this list does suggest anything like avoidance, it would be of the vii°. It never appears following anything in the list. A Stochastic chord progression based on this list would only ever produce a vii° if it started there.