I think your chances are very good. For one thing, it doesn't sound to me like you are intending to be a "rat choice" numberhead, so I wouldn't worry about your quantitative background. There are plenty of programs where more qualitative research would fit in nicely. And maybe you are a math wiz and just haven't taken the sort of courses that would apply to doing statistical poli sci research. Programs usually offer a "methods" course or course sequence so you could simply take one of those if number crunching is actually what you want to do (though I do not sense that).