91Ó°ÊÓ

High school seniors with strong academic records apply to the nation's most selective colleges in greater numbers each year. Because the number of slots remains relatively stable, some colleges reject more early applicants. The University of Pennsylvania received 2851 applications for early admission. Of this group, it admitted 1033 students early, rejected 854 outright, and deferred 964 to the regular admissions pool for further consideration. In the past, Penn has admitted about \(18 \%\) of the deferred early admission applicants during the regular admission process. Counting the additional students who were admitted during the regular admission process, the total class size was 2375 (USA Today, January 24,2001 ). Let \(E, R,\) and \(D\) represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool. a. Use the data to estimate \(P(E), P(R),\) and \(P(D)\) b. Are events \(E\) and \(D\) mutually exclusive? Find \(P(E \cap D)\) c. For the 2375 students admitted to Penn, what is the probability that a randomly selected student was accepted for early admission? d. Suppose a student applies to Penn for early admission. What is the probability the student will be admitted for early admission or be deferred and later admitted during the regular admission process?

Step by step solution

01

Calculate Total Early Applications

The total number of early admission applications is given as 2851.

02

Calculate Probabilities for Events E, R, and D

Compute \( P(E) = \frac{1033}{2851} \) since 1033 students are admitted early. Compute \( P(R) = \frac{854}{2851} \) since 854 students are rejected outright. Compute \( P(D) = \frac{964}{2851} \) since 964 students are deferred.

03

Calculate Mutual Exclusivity

Events \( E \) and \( D \) are mutually exclusive because a student cannot be both admitted early and deferred at the same time. Therefore, \( P(E \cap D) = 0 \).

04

Calculate Probability of Being Accepted Early

There are 1033 students admitted early out of a total of 2375 admitted students. Thus, \( P(\text{accepted early}) = \frac{1033}{2375} \).

05

Calculate Probability of Admission via Deferred Path

During the regular admission process, \( 18\% \) of deferred applicants are admitted. Therefore, \( 964 \times 0.18 = 173.52 \approx 174 \) additional students from the deferred pool are admitted.

06

Calculate Total Probability of Admission Through Early or Deferred Path

Using probabilities from early admission and deferred admission, calculate \( P(E \cup D') = P(E) + P(D') \) where \( D' \) is the event of being deferred and later admitted. Therefore, \( P(\text{admitted early or deferred/admitted}) = \frac{1033}{2851} + \frac{174}{2851} = \frac{1207}{2851} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Early Admissions

Early admissions refer to the process by which students apply and receive admission decisions from a college before the standard notification period. This practice allows students to secure a spot at their preferred institutions ahead of the regular admissions cycle. Applicants who opt for early admission typically submit their applications by November, well before the regular decision deadlines.
Benefits of early admissions include:

• Reduced stress from the college admission process
• More time to focus on final-year studies and extracurricular activities
• The potential advantage in admission rates, as some colleges prefer early applicants

However, the early admissions process can be competitive due to limited slots, and applicants should ensure they meet all criteria and deadlines to improve their chances of acceptance.

Mutually Exclusive Events

Mutually exclusive events are events that cannot happen at the same time. In the context of the admissions process, when a student either gets admitted early or deferred, these events are mutually exclusive.
This means:

• A student admitted early cannot also be deferred.
• Therefore, the probability of both events occurring simultaneously, denoted by \(P(E \cap D)\), is 0.

Understanding mutually exclusive events helps in accurately calculating probabilities without overlapping instances, which is crucial in statistical analysis, such as determining admission probabilities.

Deferred Admissions

Deferred admissions refer to applicants who are not rejected outright or admitted early but are placed in the regular admissions pool for further review. This provides an opportunity to reassess their application alongside regular applicants at a later date.
Key features of deferred admissions include:

• Deferred applicants have their files reviewed again during the regular decision phase.
• There is a chance of acceptance during regular admissions, often lower than the initial early acceptance rate.
• At the University of Pennsylvania, for example, approximately 18% of deferred applicants were eventually admitted during the regular decision round.

Deferred admissions can be seen as a second chance, offering students hope for acceptance later on, even if they weren't accepted early.

Admission Statistics

Admission statistics are vital in understanding the overall competitiveness of college admissions processes. Data points like the number of applicants, acceptance rates, and deferred applicants offer insights into the likelihood of gaining admission.
In this case:

• A total of 2851 early admission applications were submitted.
• Of these, 1033 were admitted, 854 were rejected, and 964 were deferred.
• The probability \(P(E)\) of an applicant being admitted early is \(\frac{1033}{2851}\).
• The probability \(P(\text{admitted early or deferred/admitted})\) combines both early and deferred admissions, calculated using \(\frac{1207}{2851}\).

Statistical analysis of such data helps students and educators understand the dynamics of admissions better, enabling them to prepare more effectively for the application process.