Chance of At Least One Acceptance: The Math Behind a Balanced School List

The Formula

If an applicant has an admission chance of p at a school, the chance of being rejected there is 1 − p. The chance of being rejected at every school on a list is the product of those rejection chances, and the chance of at least one acceptance is one minus that product.

Chance of at least one acceptance = 1 − (1 − p₁) × (1 − p₂) × … × (1 − p_n)

A Worked Example

This list uses the same applicant as our school-list sorting guide: a 164 LSAT and a 3.70 GPA, scored against five law schools' 2025 ABA 509 figures. The modelled chances come from AdmitBase's match engine.

After all five schools the chance of no offer is 13.9%, so the chance of at least one is 86.1%.

Where the Points Come From

Remove one school at a time and see what the total does:

The Safety is worth about 28 points. The Far Reach is worth less than one. That asymmetry is the whole argument for balancing a list instead of lengthening it. Adding three more Far Reaches at 4% each to the four reaches takes that list from 57.7% to 62.6%, an improvement of about five points for three more application fees. Adding the one Safety takes it to 86.1%.

What Adding Schools Does When They Are All Alike

If every school on a list had the same admission chance and the decisions were independent, the chance of at least one acceptance would climb like this:

Twenty applications at 5% each reach only 64%. Five at 30% each reach 83%. The tilt of the list towards schools where the applicant is competitive does more than the length of the list.

The Caveat: Decisions Are Not Independent

The formula treats each school's decision as a separate coin flip. They are not. The same transcript, test score, personal statement and letters go to every school, so an application that is weaker than its numbers suggest is weaker everywhere. That correlation lowers the chance of at least one acceptance relative to the independent calculation.

A simple illustration shows the direction and rough size. Suppose half of all years are "good" for the applicant, with every chance 50% above its modelled value, and half are "bad", with every chance 50% below it, so the average is unchanged. With a 10% modelled chance at each of 20 schools, the independent formula gives 88%; the good-year, bad-year version gives about 80%. This is an illustration built for this article, not a measured figure, and the true correlation between schools is unknown. The practical lesson stands: treat the independent number as an upper estimate, and use it to compare lists, not to promise an outcome.

What to Tell an Advisee

• Lead with the mix, not the count. The question is how many schools are genuine safeties, not how many applications are in the cart.
• Show the number move. Have the advisee tick schools and watch the total change; they will see for themselves that a long list of Far Reaches barely moves it.
• Say what the number is. A modelled estimate that assumes independence, based on published medians, not a probability of admission to any one school.
• Remember the other costs. Every extra application has a fee and a secondary or supplement. Past the point where the total stops moving, the list is getting more expensive, not safer.

Try It on a Real List

The law school chances calculator and its siblings for medicine, dentistry, pharmacy, veterinary medicine, optometry and MBA programs have a free Build your list panel: tick the schools an applicant is considering and the chance of at least one acceptance, and the chance of none, update with each one. The full Monte Carlo simulation of a list, run over 10,000 simulated application cycles, and the names of the schools that would raise the total most, are part of the Cycle Report.