No, NROTC and AFROTC place a heavy focus on STEM majors. They need nuclear engineers not necessarily political science majors(me). AROTC doesn't heavily factor in your major unless you are premed or nursing and they have a separate applicant pool. It is just as competitive though. At the end of the day, you need to go into the branch of the military that best fits you, regardless of the scholarship. Don't go into it just hunting for a scholarship. The money is the bonus. At least that's the way I look at it.
Each branch has different minimum requirements to apply and differences in the average scores of accepted candidates. So, the chances of winning are not the same among the branches. As far as Army, you can list up to seven college choices (some must be in-state public schools) on your application and there is a winner can be awarded scholarship offers to use at up to three schools on the list. Which schools the winner received scholarships for depends on several factors - I don't think tuition cost is one of them, as far as Army goes. It depends on how many other scholarship winners applied, how many scholarship are allocated, and sometimes, choice of major.