You’re new here, so possibly you haven’t seen this thread: http://talk.collegeconfidential.com/massachusetts-institute-technology/939227-reminder-no-one-not-even-me-can-give-you-an-accurate-chance-at-mit-p1.html. It’s about MIT, but holds true as well for Princeton.
You sound like a wonderful candidate, as will the vast majority of students who are rejected. Recommendations and essays will probably determine the outcome, and we are not the people reading those.
As you also posted a similar chance thread on the Harvard forum, you should read this thread: http://talk.collegeconfidential.com/harvard-university/1420290-chance-threads-please-read-before-posting-one-p1.html
Best of luck to you!