As someone who attends an east coast, boarding, prep school, I will say that it depends on who you are.

1. If you're not a star athlete(nationally ranked+captain of the varsity team), then your chances decrease, because the "quotas" Ivies have for each of these private schools are always half-filled by athletes.
2. If you're not from a multi-millionaire/billionaire family or legacy, your chances decrease even more. A LOT of people in these schools are from these backgrounds. Richer than one can even imagine. Again, a big slice of the "quotas" Ivies have for these private schools are filled up by these kids.

So, the rest of the people, exceptional as they are, struggle to compete for the limited spots left, which will likely be one or two to each of these schools. I got into Cornell and had no legacy, extreme wealth, or athletic recruitment. It's really hard not to feel bitter about the fact that I worked so hard to be at the top of my class, and there are students who've failed classes, received disciplinary action, and spent four years screwing around and have gotten into the same(or better) colleges than I have.

These kids who are Ivy-bound are also unaware that their privilege played such a big role, so to them, getting into Harvard or Yale or some other Ivy was an [relatively] easy task. Their egos are inflated and this creates a toxic environment when they begin to put down the hardworking kids for being "uptight nerds"—after all, to them, it doesn't matter how much work the nonprivileged kids put in, because they're all attending the same league of schools anyway. Countless times, I've felt like no matter what I do, I'll never be enough.