Well, SpaceX definitely has the advantage of a serious head start. However, it really depends on what you mean by "private space race." If you are talking about commercial space tourism, one could make the argument that Blue Origin has focused pretty much exclusively on that target, while SpaceX has focused on building and launching advanced rocket technology. Now, if SpaceX is successful with its crewed capsule that is in the works for NASA, they will have demonstrated that they can launch people, which will lead to space tourism sponsored by Elon Musk. However, remember that Boeing is doing the same thing for NASA. True, Blue Origin really has not done anything but launch (and land) rockets, but I would not be surprised if Bezos got a commercial space tourism contract before Musk.