If all 8000 applicants applied to all the schools (4000 positions), then the acceptance rate would be ~50%.
However, some applicants only apply to a couple of schools. Others apply to 20+ schools.
For example, let's take hypothetical applicant #220. For theoretical reasons, let's assume that schools #32, 37, and 48 would be more than willing to give applicant #220 an acceptance. Applicant #220 only applied to 8 schools, however and none of the schools (#1-#8) gave him an interview or an acceptance.
His personal statistics will show that his success (so to speak) was 0/8, or he had zero acceptances out of 8 schools. Had he applied to schools #32,37, and 48, he would have had three acceptances. If he had substituted those schools in place of some of the eight he did apply to, his acceptance rate would be 3/8. Likewise, if he merely added those schools to his original application list, his ratio would be 3/11. And certainly, his ratio could be 3/54 if he had applied to all of the available schools.
Our point is that although there are seats available for 50% of applicants, the statement that "roughly half the applicants" will get accepted somewhere is not true unless ALL applicants apply to ALL schools.
What about the guy who applies to 7 schools and gets in at all of them? He has a 100% acceptance rate. The guy who applies to 19 and gets in nowhere has a 0% success rate.