The town I grew up in was very polarizing. We had both extreme wealth and extreme poor. As a retrospective, we had very equal opportunities from all income brackets. There were wealthy kids who had huge success but many others who fizzled out and didn't do anything. Some ended up with extensive criminal records.
On the opposite end of the spectrum, some of the extremely poor kids I grew up with could barely afford ramen as a family. Some ended up drug dealing and in jail, others ended up with massive success.
There were a few key attributes that I think determined success vs. failure for all wealth brackets. We had some very key teachers in shop classes who encouraged folks who didn't think they could go to college to specialize in trade skills. Those who took it seriously and worked hard at it are largely stable or well off. Those who didn't fizzled out.
Obviously this is all anecdotal as it was one graduating class in one city in one state. However, in my own experience the desire to work hard dictates much of the success of an individual. It would be foolish to say that it is the ONLY thing contributing to success but I believe that it is a large portion of it.
Also, that word "lucky" drives me nuts but I'll save that for another time :)
Determination and readiness to work hard enable someone to take advantage of an opportunity. But you still need the opportunity.