You may be interested in https://www.nytimes.com/interactive/projects/college-mobilit... which compares Harvard against "other Ivy league schools" on a typical / above / below average. Looks like just under 5% is typical for bottom-20%-income background for admitted students.

It's also worth noting that Harvard, for example, admits fewer than 3,000 students per year. That means that 100-150 students (out of 40,000 applicants) will come from a poor background.

Seems like being rejected from an Ivy league school is statistically the most likely outcome, regardless of what your qualifications are.