How many colleges should I apply to?
If I apply to six schools, what are my chances of getting into at least one? Many websites (e.g. here, here, here and here) provide calculators that tell you your chances of getting into a specific school given your SAT scores and GPA. That is super-useful, but it does not tell you the chance of getting into at least one or at least two of the schools in the set of schools that you applied to. For example, if you apply to three Ivy League schools and your chance at each is 5%, what are the chances that you will get into at least one?
Anyone applying to an Ivy League should be able to calculate these odds using their phone — just raise 0.95 to the third power and subtract the result from one, you get 14%. What are your chances of getting into at least two? Well, that is a bit trickier, but a binomial probability calculator can handle this. The answer is less than 1%. (The number of successes from n trials with a p probability of success for each trial is distributed Binomial.)
In reality most students apply to schools with different chances of getting in — some “reaches”, some “likelies” and some “safeties.” If you apply to three schools with a 5% chance, three schools with a 20% chance and three schools with a 50% chance, what are the odds of getting into at least one? That is a wicked challenging problem. When we have multiple trials but the chance of success at each trial is different, the number of successes follows the Poisson Binomial Distribution. It takes a lot of computing power and some sophisticated algorithms to get those probabilities. Just consider the fact that if you apply to 20 schools, there are 184,756 different combinations of getting into 10 of them.
Humans are not very good at estimating probabilities. In some contexts, e.g. finance, we tend to underestimate rare events. In other contexts, e.g. plane crashes and earthquakes, we tend to overestimate their likelihood. We are also terrible at calculating exponentials. The purpose of this article is to help high school students and their parents develop some intuition regarding their chances of getting into x number of colleges when applying to n of them. Most importantly, the article provides a quantitative assessment of common rules of thumb strategies of applying to college.
Applying to college is a high-stakes game. Playing the game without knowing the odds of different outcomes seems imprudent.
The chart above shows the effect of adding less selective schools to one’s portfolio of applications. The red line shows the chances of getting into at least x number of schools when you apply to only highly selective schools — three schools with chances of 5% each. Adding three schools with a 20% chance each raises the probability of getting into at least one school from 14% to 56%; and adding three more schools with a 50% chance of getting into raises that probability to 95%. It is worth pointing out how quickly the chances of getting into more than one school fall. For example, the green line shows the probabilities when applying to nine schools — three with 5%, three with 20% and three with 50% chances. With this mix of schools, the probability of getting one or more is 95%, but the probability of getting into three or more is only 40%.
The above calculations depend upon a reasonably accurate assessment of one’s chances of getting into each school. These chances are based on each school’s acceptance rate and how the student’s high school record compares to the profile of accepted students. Websites such as PrepScholar, Niche or CollegeVine and many others provide these assessments for individual schools. Also, one crucial assumption is that the events of getting into schools are independent. While there is a great deal of noise in admission decisions, it is quite possible that decisions are correlated across schools — if one committee likes certain aspects of your application, chances are that other committees will too, and vice versa.
If you would like know your chances of getting into x schools given your own set of n schools, I built a calculator that allows you to customize your list of schools. Below is a screenshot and you can access it here.
