This advice comes with the caveat that I went through the selection process several years ago, but I think things have been fairly stable throughout that time.
For most applicants, I think the magic number for 1 or more acceptances is somewhere between 3-5. This is highly dependent on if you're a good applicant and what kind of programs you're interviewing at. If you are an average applicant and are interviewing at top 10 programs, it will be higher. If you are an awesome applicant interviewing at top 11-25 programs, it will be lower.
KPCOFGS is correct in noting that interviewees and applicants are very different pools though. There may be a large number of applicants who simply don't meet basic criteria for admission. They may get only 1 or even no interviews. Most of the people garnering 3+ interviews are good enough to get in somewhere and ultimately will.
My guess is that a typical NIH MSTP program
interviews: 8-10 per spot
initially accepts (understanding that some will decline): 1-2 per spot
ultimately accepts (including waiting lists and such): 3-5 per spot
So, once you make it to the interview stage I think your chances are anywhere from 30-60% at that program. You're really helped out by the fact that the pool of truly qualified applicants is not that big and is pretty self selecting. The really good applicants may be interviewing from 5-12 places and will probably have to turn down 1 or more offers, which is why many places end up accepting so many people per spot.
You can argue that this won't be true at the very best programs (Harvard, Penn, Hopkins, UCSF, etc), but there is still some truth to it because the same people are interviewing at multiple top programs. They can't attend more than 1!