It is NOT completely random. Only partially random.
The champion you roll is based on what you have available. Free Champions in addition to whatever you actually own. Someone already supplied the math on that; so, I won't copy pasta it in my response.
Then you have the re-roll mechanic that allows players who have completed an ARAM match to reroll a champion they do not wish to play.
Champions that are re-rolled are then up for grabs to any ally.
Additionally, players are able to trade champions.
Players may also queue up with friends whose champion pool is identical to theirs so as to increase the odds of them rolling their preferred champions.
In summation, the randomness of ARAM is not absolute. Players have quite a bit of control over the champion they may play simply by having their owned champion pool reduced.
This is why ARAM only accounts exist and is why players have repeatedly suggested to make ARAM a truly random experience by having all champions unlocked for all players regardless of what they do or do not own. I suspect Riot refuses to do this as it would likely upset the players who have meticulously crafted ARAM only accounts through smart purchases. Players who have likely bought skins for their ARAM champions. And Riot is unlikely to make any decision to improve the state of the game if the decision is liable to threaten their sales.