OTHER WORD CHOICES: chance, accidental, haphazard, unsystematic, hit and miss
AND yes, arbitrary is another word for Random.
Here are two off the wall words that I found that also can be synonyms for Random:
Stochastic: (stuh-KAS-tik) adjective: Involving chance; random; probabilistic.
Etymology: From Greek stokhos (aim, target, a pointed stake for an archer to aim at). Earliest documented use: 1662.
Usage: “Medicine is a stochastic science — no doctor can predict the future.” Sandeep Jauhar; When Doctors Slam the Door; The New York Times; Mar 16, 2003.