Question

For high school students graduating in 2007 , college admissions to the nation's most selective schools were the most competitive in memory. (The New York Times, "A Great Year for Ivy League Schools, but Not So Good for Applicants to Them," April 4,2007 ). Harvard accepted about \(9 \%\) of its applicants, Stanford \(10 \%\), and Penn \(16 \%\). Jorge has applied to all three. Assuming that he's a typical applicant, he figures that his chances of getting into both Harvard and Stanford must be about \(0.9 \%\). a) How has he arrived at this conclusion? b) What additional assumption is he making? c) Do you agree with his conclusion?

Short answer

Jorge used independent probabilities, but his assumption of independence might be unrealistic, making his conclusion potentially inaccurate.

Step-by-step solution

Understanding Probability Calculations

Jorge calculates the probability of getting into both Harvard and Stanford by multiplying individual probabilities. Harvard's acceptance rate is \(9\%\) or \(0.09\) and Stanford's is \(10\%\) or \(0.10\). Thus, \(0.09 \times 0.10 = 0.009\), which is equivalent to \(0.9\%\). This multiplication is based on the assumption the events are independent.

Identifying Assumptions

Jorge assumes that his chances of being accepted to both institutions are independent, meaning acceptance to one does not affect the probability of acceptance to the other. This leads him to calculate the probability as the product of the individual probabilities.

Evaluating the Conclusion

Jorge's assumption of independence may not hold true in reality. Colleges can share information between them, or the student's profile may receive similar evaluations by both. This dependence means Jorge's calculation might not be accurate, as the events likely are not independent.

Key concepts

College Admissions

College admissions, especially to top-tier universities, can be an intense and competitive process. Each college sets its criteria to decide which students get accepted. Selecting schools like Harvard, Stanford, and Penn have acceptance rates that are often in the single digits. These prestigious institutions evaluate numerous applications for a limited number of spots. The process requires students to submit comprehensive applications showcasing their academic achievements, extra-curricular activities, and personal essays. Admission officers review each application extensively, seeking not just academic excellence, but also diverse backgrounds and unique talents. Understand that certain factors, like legacy status or sports recruitment, can influence an applicant's chances significantly. This makes predicting one's acceptance more intricate than just looking at quantitative measures alone. College admissions are ultimately a blend of art and science, with many moving parts, including grades, test scores, and more subjective factors like recommendations and interview impressions.

Independent Events

In probability theory, the concept of independent events is crucial. Two events are considered independent if the outcome of one does not affect the outcome of the other. An analogy is flipping a coin twice. The result of the first flip does not impact the second. When Jorge multiplies the individual probabilities of acceptance to Harvard and Stanford, he assumes these events are independent. This implies the probability of being accepted to one does not influence the chances of being accepted to the other. However, this independence assumption might not always be realistic in college admissions. It is essential to realize that several interconnected factors could lead the events to be dependent. For example, universities may evaluate similar aspects of a student's application, or share information about applicants. Thus, making the assumption that college admissions are entirely independent events might oversimplify the probability calculation and lead to misleading outcomes.

Acceptance Rate

The acceptance rate of a college is a metric that indicates the percentage of applicants that are admitted over a given admissions cycle. Calculated as the number of acceptances divided by the number of total applications received, it serves as a general measure of a school's selectivity. For example, if Harvard receives 50,000 applications and accepts 4,500, its acceptance rate would be 9%. Acceptance rates can be indicative of a school's prestige and competitiveness, with elite institutions generally having lower acceptance rates. However, a low acceptance rate can also result from factors other than selectivity, such as a large number of applications driven by a school's popularity or expansive applicant outreach. It's important for students to not only consider acceptance rate when applying to colleges, but also how well each institution aligns with their academic and personal goals. Therefore, while acceptance rates are an important factor, they are just one piece of the larger admissions puzzle students should evaluate.