Calculating the Probability of College Admissions

Here’s a fun math problem:

 If a college applicant applies to 20 schools, each with a 5 percent admission rate, what is the probability that she will be accepted to at least one of those schools? 

Is it 5 percent? 100 percent?

Hint: It’s not 100 percent, but that’s what many students believe, an example of what I call “magical thinking.” (Others may call it “not knowing how probability works.”) Actually, the answer is around 64 percent!

Now, you may be thinking: You’re telling me that if I apply to 20 schools with admission rates the equivalent of Stanford’s, I have a 64 percent chance of getting into at least one of them? Unfortunately, no, that’s not correct either.

This word problem above assumes that the process of college admissions relies entirely on random chance, like that of rolling dice, but that’s certainly not how college admissions works. A student’s qualifications (e.g., grades, scores), demographics (e.g., gender, race, geographic location) and other application components (e.g., essays, letters of recommendation) are the determinants of admission for every applicant. This means that, for many students – in fact for the vast majority of students –  the chance of being admitted to at least one of the schools in my word problem scenario may be virtually zero  . That’s because the vast majority of students are, by definition, not at the top of the curve. If you apply to a college for which you’re not academically competitive, you actually have a significantly lower probability of being admitted than a qualified student does, unless you fall into a special group such as recruited athletes, legacies, and under-represented minorities.

Failing to have a realistic understanding of one’s probability for college admission is what I call magical thinking. Why? Because it’s the same kind of thinking some people use when they buy multiple lottery tickets. Math goes out the window, and blind, irrational hope takes over.

 Now here is some good news about the numbers: that 5 percent admission rate in all likelihood isn’t really 5 percent.  In fact, the reported number may be significantly and, in some ways, intentionally deflated.

It is true that the number of applications is growing, and here are a few reasons why:

• The population of the United States continues to grow, but the number of seats at colleges and universities has either remained unchanged or has grown at a much slower rate.
• The portion of the United Stations population that is college-bound has significantly increased.
• Thanks to the Common Application, the Universal College Application, and the new Coalition Application, it’s easier to apply, and so students are applying to more colleges.
• More families have access to the Internet, making it easier to find out about colleges that aren’t close to home.
• There are more international applicants than ever before, adding to the competition for seats.

So, higher numbers of applications are part of the mathematical story. More demand for the same number of seats results in a lower acceptance rate, which is scary for applicants but delights colleges. Lower acceptance rates make the colleges look better, so their goal is to try to improve that number.

This is how they do it: Colleges play numbers games to increase their rankings. For example, some colleges have been found to count incomplete applications when reporting their total number of apps received. Since those applications are by default denied, the percent of students admitted drops.