Posted by: W. E. Poplaski | July 31, 2008

## College Applications: How Many?

You have your list of colleges; schools to which you have a decent shot at gaining admission (i.e., your SAT or ACT scores match those of their freshman classes, as does your GPA and high school class rank).  The trouble is your list has 30 colleges!  At about \$50 per application, that’s \$1500!  You need to shorten the list, but by how much?  Obviously, it would be risky if you only applied to one school—what if you were rejected?

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Binomial expansion can help you. It is a great tool for answering questions such as, “to how many schools should I apply to be 90% confident (or any other percent) of being accepted by at least one school (or two schools, or three schools, or…)?”

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Why and how does one do binomial expansion?  It’s as easy as A, B, C.

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Part A.

The binomial of interest looks like this: (a + r)n, where a is the fraction of applicants accepted, r is the fraction of applicants rejected, and the exponent, n, is the number of schools to which you apply.

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Expanding a binomial just means, “multiply the binomial by itself n times.”  The expanded binomial will be a bunch of terms connected together by ‘+’ signs.  In fact, the number of terms will be one more than the number of schools to which you apply (e.g., if you apply to 8 schools, then the expanded binomial will have 9 terms).

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Each term of the expanded binomial represents the probability of being accepted by a particular number of schools.  Let’s continue with the example of eight schools.  The first term is the probability of being accepted at all 8 schools.  The second term is the probability of being accepted at 7 of the 8 schools. Each succeeding term is the probability of being accepted at 6, 5, 4, 3, 2, 1, and finally, 0 schools.

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Part B.

What you need is the probability associated with the last term, getting accepted at zero schools.  Why?

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Let’s look at an example…if the probability of getting accepted at 0 out of 8 schools happens to be 10%, then the probability of getting accepted into at least one school (i.e., 1, 2, 3, 4, 5, 6, 7 or 8 schools) would be 90% because 100% –10% = 90%.  All you need to do is subtract the probability of that expanded binomial’s last term from 100% to get your answer.

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Part C.

Now play with expanding a binomial to see what’s going on—hardly any math skills are needed! If you remember this stuff from math, great, otherwise don’t worry.  You just need the mechanics of the process, which you can get from observing the patterns.

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Expanding the binomial means to multiply the binomial by itself n times.  There are cheat tricks you can use for help, for example:

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(a + r)1 =  1a1  +  1r1

(a + r)21a2   +  2a1r1  +  1r2

(a + r)31a3   +  3a2r1  +  3a1r2  +    1r3

(a + r)41a4   +   4a3r1 +   6a2r2   +   4a1r+ 1r4

(a + r)51a5   +  5a4r1  +  10a3r2  + 10a2r3  +  5a1r4 +  1r5

(a + r)61a6   +  6a5r1  + . . .                                         6a1r5 +  1r6.

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A few important observations to make about expanded binomials:

1. The number of terms in the expanded binomial (the stuff on the right-hand side of the ‘equals’ sign) is one more than the number of colleges to which you are applying (which is n, the exponent on the left-hand side of the ‘equals’ sign).  For example, if you are applying to three colleges then you need four terms in the expanded binomial.
2. Each term of an expanded binomial consists of a and r multiplied together, except the first and last terms.
3. The sum of the a and r exponents in each term is equal to n. Assigning exponents to each variable of the terms in the expanded binomial is easy if you notice the pattern they follow.
4. The coefficients (the red numbers), likewise, follow a pattern.  Pascal’s pyramid is an easy way to find these coefficients (this website explains Pascal’s Triangle: http://en.wikipedia.org/wiki/Pascal’s_triangle ). Simply count the number of terms in your expanded binomial—that’s the number of coefficients you need.  Then find the row in Pascal’s triangle that has that number of coefficients.  (Here’s a website that generates Pascal’s Triangles for you: http://mathforum.org/dr.cgi/pascal.cgi?rows=12.)  Copy those coefficients—in the order they appear in Pascal’s triangle—into your expanded binomial, one coefficient per term.  Notice that the first and last terms of the expanded binomial always have ‘1’ as their coefficient.

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This website provides details on the mathematics of expanding binomials:

http://www.regentsprep.org/Regents/math/algtrig/ATP4/bintheorem.htm

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Now, you are ready to see how to use this tool.  Suppose you apply to 6 colleges.  Use binomial expansion to find the probability of getting into at least one college.  Your expanded binomial looks like this—

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(a + r)6 = 1a6 +  6a5r1 + 15a4r2 + 20 a3r3 + 15 a2r4 + 6a1r5 + 1r6

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Each term represents a different outcome in the number of acceptances and rejections you can get from applications to six colleges (in fact, all possible outcomes are listed by the expanded binomial, from six acceptances through six rejections).  For example, the third term, 15a4r2, represents the probability of four acceptances and two rejections—you can tell from the exponents.  The sixth term, 6a1r5, represents one acceptance and five rejections.

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Here’s where the selectivity comes in.  A selective college might have a ≈ ¼ and r ≈ ¾, meaning that they accept approx. 25% of their applicants and reject 75%. (this website explains selectivity: http://www.collegedata.com/cs/content/content_choosearticle_tmpl.jhtml?articleId=10004 )

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So, if you want to know the probability of getting accepted into at least one college—from a list of six colleges each having a rejection rate of 75%—you would first calculate the probability of getting no acceptances (trust me, you will see why shortly): 1r6= (¾)6 =   0.18.  Then, you would subtract that number from 1.00 (that is, 1.00 – 0.18= 0.82).  Your probability of being accepted by at least one of the six colleges is 82%.

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How does that calculation work?  Each term represents the probability of being accepted by a particular number of schools.  The total probability—obtained by summing the probabilities of all terms in the expanded binomial—is 1.00 or 100%.  You are interested in finding the probability of at least one acceptance; that is the same as the sum of the probabilities for 6, 5, 4, 3, 2, and 1 acceptances.  However, that is also the same as subtracting the probability of 0 acceptances from the total probability of 100%.

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[Side note:  What if you wanted the probability of being accepted by, say, exactly two out of six colleges with this selectivity?  It would be: 15a2r4 or 15(¼)2(¾)4 = 15(0.0625)(0.3164) = 0.31 = 31%.  Also, being accepted at exactly 1 out of 6 colleges is 6a1r5 = 6(0.25)(0.24) = 0.36 = 36%, which is different than being accepted by at least 1 out of 6 colleges.]

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Applying to six schools gave you a probability of 82%; what about applying to seven schools?

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Remember, we only need the last term of the expanded binomial—acceptance at 0 schools—and that is 1r7= (¾)7 =0.13; giving us 1.00 –0.13 =0.87 or 87% for seven colleges.

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And for eight colleges…the term we need is 1r8, which is (¾)8 =0.10; giving us 1.00 –0.10 =0.90 or 90%.  In other words, for schools with that level of selectivity—and if your credentials match those of their student body—you should apply to eight schools to be 90% confident of getting at least one acceptance. (Of course, an 82% probability of acceptance is pretty darn good, too, and costs \$100 less than having a 90% probability).

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If you study the binomial expansion you will see that for less selective colleges you can apply to fewer of them and still have a high probability of getting accepted into at least one.

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So, to answer the question, “to how many schools should you apply?”, you need to consider how confident (i.e., what probability of acceptance) you want to be.  Of course, high probabilities are associated with a greater number of school applications and higher costs.

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…to how many schools should you apply (with a = ¼, r = ¾) to be at least 99% confident of being accepted by at least one school?  We can do some algebraic manipulation to isolate and then solve for n.

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r n ≤ 0.01

n(log(r)) ≤  -log(0.01)

n ≥ -log(0.01)/(-log(r))

n ≥ -log(0.01)/(-log(0.75)) > 16.

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You must apply to more than 16 schools. Ouch, that’s more than \$800!

[Remember, to be at least 99% confident, the term 1rn must be ≤ 0.01, because 1.00 – 0.01 = 0.99…so, if you wanted to be at least 95% confident, then 1rn ≤ 0.05]

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## Responses

1. Youch, \$800 is a lot, but 99% sure is always rather nice.

Personally, I’m all for applying to a low school as a backup and then taking my chances with the others 🙂

2. Sounds great, but then, to how many others would you apply? …or, perhaps only apply to a ‘low school’?

If you only get into your ‘low-ball’ school, are you prepared to get involved in a lot of extra-curricular activities to keep yourself challenged (or go through the process again and then transfer)?

When you graduate from college, will your high GPA and the experiences you get put you in a position to compete in the market-place against those who graduate from a more competitive college?

I think the answers to those questions vary with the student involved. Some students can successfully manage their education so that they have meaningful experiences. However, many students don’t know what is valued in the world beyond college and rely on the college to provide that for them.

The point is to understand the situation you are in—as best as possible—and make decisions based on that understanding.

Surely, applying to 20 schools is not a cost effective solution for most students, and that is the kind of thing binomial expansion can tell you.