In addition to Centuros' comment, there's been some anecdotal evidence that if you own a champ on the Free Roster, you have a higher chance to roll them (as if they're on your list twice).
There's also the bias beyond just ARAM only accounts. My friend and I both play ARAM often, but as I only unlock champions I know I can handle and play well, my pool is smaller and I'm more reliable on them all. He however, is out to get every champion, and thus his pool is larger/less reliable, making him less likely to want to ARAM. He regularly trades in lobbies, and spams his rerolls when they're available. I've gone on 30 game streaks without having to reroll.