I get the feeling that ARAM isn't truly random. For one, it seems that free champs are present in games more often than others, and pretty often I go through streaks of rolling the same champion. Now I suppose, these could be statistical anomalies and the free champs probably have high likelihood due to being available to all, but I'm curious if Riot has said anything about this.
To provide some quasi-data, I've played about 500 ARAMs and I own about 25 champions and there are two or three that only rarely get rolled. Now this is probably a biased recounting, but I can't help but wonder.