Robert Downey Jr.'s son being arrested is not surprising. Robert Downey Jr.'s legal problems have played out in the public eye and in front of his children. He has shown through his actions that it is okay to break the law. By providing this type of role model for his son, he has set him up for failure.
I believe that most children learn by example. For the majority of the population, that example is mom/dad. Parents have the unique job of forming their children into the kind of adults that they would want them to be. Unfortunately, parents are human too. Much too often, parents get caught up in their everyday lives that they forget they have a set of eyes and ears watching their every move.
Yes, Robert Downey Jr.'s son is apparently just like his father, and not in a good way. Downey's son has no doubt been influenced by his father's many escapades and it is no surprise that he has been arrested. This arrest will probably be the first of many, following in his father's footsteps.
I do not believe that the fact that Robert Downey Jr.'s son was caught in possession of drugs necessarily means that he is following in his father's foot steps of drug-addiction. It may just be a matter of him being young and experimenting with illegal drugs. In my opinion, if someone gets caught with drugs, they are not necessarily an addict.