Random is a concept that means it can't be predicted and that everyone has an equal chance being in any of the five positions, but it's kind of difficult to make a system that is truly not dependent on any factors. The best we can do when it comes to electronic/virtual randomness is impose an algorithm or formula that mimics the effect of randomness as best as it can.
If this kind of stuff really gets you going, then realize that in 2015 we've come up with some pretty mind blowing algorithms that simulate randomness almost indecipherably, but this is Riot we're talking about. Not to intentionally pick on them, because I know they get a lot of fuss, but at this point when you start to realize that maybe they just grabbed any algorithm, said "good enough," and slapped it in their game... it's probably true. I've had issues with their "random" generators. When you own all of the champions but ARAM selects Mordekaiser for you 9 games in a row before selecting Mundo, and then Mordekaiser a handful more times, and then two more Mundos... while definitely 100% possible in terms of randomness.. probably safe to bet their random algorithm could use some work. I've started typing it as A"R"AM to my friends.
I also get selected to be in the same position for several games in a row, but when the chance is 1/5, it's more believable that it would happen "randomly."