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Chapter 4: Problem 29

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. Suppose that for a recent admissions class, an Ivy League college received 2851 applications for early admission. Of this group, it admitted 1033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration. In the past, this school has admitted \(18 \%\) of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was \(2375 .\) 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 who were admitted, what is the probability that a randomly selected student was accepted during early admission? d. Suppose a student applies for early admission. What is the probability that the student will be admitted for early admission or be deferred and later admitted during the regular admission process?

Short Answer

Expert verified
a. \( P(E) = 0.362, P(R) = 0.299, P(D) = 0.338 \). b. Yes, they are mutually exclusive, \( P(E \cap D) = 0 \). c. \( P = 0.435 \). d. Calculate \( P(E \cup D_{admitted}) \).

Step by step solution

01

Understanding the given data

We have three events from the early applications: admitted early (1033), rejected (854), and deferred (964). The total number of early applications is 2851. There are 2375 total students admitted at the end, after considering both early and regular admissions.
02

Calculate Probability of Each Event (a)

To calculate probabilities, we divide the number of favorable outcomes for each event by the total number of early applications (2851). \( P(E) = \frac{1033}{2851}, \quad P(R) = \frac{854}{2851}, \quad P(D) = \frac{964}{2851} \). Calculate these probabilities:
03

Determine if Events are Mutually Exclusive (b)

Events \(E\) and \(D\) are mutually exclusive if they cannot happen at the same time. Since a student cannot be both admitted early and deferred, \(P(E \cap D) = 0\).
04

Calculate Probability of Early Admission among Admitted (c)

We need the probability that a student is accepted during early admission out of the total admitted students. Since 1033 students were admitted early, \(P(E \mid \text{admitted}) = \frac{1033}{2375}\).
05

Calculate Probability of Early Admission or Deferred and Later Admitted (d)

We find the probability of being admitted early \(P(E)\) and the probability of being deferred and then admitted. Out of the deferred, 18% are admitted later: \(0.18 \times 964\). Calculating this gives the total probability: \( P(E \cup D_{admitted}) = P(E) + P(D_{admitted}) = \frac{1033}{2851} + \frac{0.18 \times 964}{2851} \).

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Key Concepts

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

Mutually Exclusive Events
Understanding mutually exclusive events is crucial in probability. These events cannot occur at the same time. For example, when flipping a coin, getting heads and tails simultaneously is impossible.
In the given exercise, early admission (event \(E\)) and getting deferred (event \(D\)) are mutually exclusive. This means a student cannot be admitted early and deferred at the same time.
Mathematically, two events \(A\) and \(B\) are mutually exclusive if \(P(A \cap B) = 0\).
  • In our scenario: \(P(E \cap D) = 0\) because an applicant cannot be both admitted early and deferred.
This concept helps simplify probability calculations as it allows us to consider probabilities individually without overlapping.
Early Admission
Early admission is a critical process for high school seniors aiming for selective colleges. It allows students to apply earlier in the academic year, typically with a higher chance of acceptance.
In the context of the exercise, early admission includes 1033 students admitted from a total of 2851 applicants. The probability that an applicant is admitted early is determined by dividing the number admitted early by the total applicants: \(P(E) = \frac{1033}{2851}\).
This process often binds the student to attend the college if accepted, making it different from the regular admission timeline.
It can significantly impact a student's college application strategy, heavily influencing their chances of attending their preferred schools.
Deferred Students
Deferred students are those not accepted immediately during the early admission phase but are not outright rejected either. They are reconsidered during the regular admission round.
In the given exercise, 964 students were deferred. Out of these, the historical data indicates that about 18% of deferred students are eventually accepted.
  • Calculating deferred admissions: \(0.18 \times 964\).
  • This means about 173 students are expected to be admitted later.
Deferrals provide students a second chance, allowing them to strengthen their application in hopes of regular admission.
Admission Statistics
Admission statistics give a comprehensive view of the admission process in selective colleges. These statistics include data on admits during early and regular admissions, deferrals, rejections, and overall acceptance rates.
The given exercise provides several statistics:
  • Total early applications: 2851
  • Admitted early: 1033, which is approximately \(36.2\%\)
  • Outright rejected: 854
  • Deferred to regular: 964
  • Total class size: 2375
Combining these, the probability that a randomly selected student among those admitted was accepted during early admission is \(P(E \mid \text{admitted}) = \frac{1033}{2375}\).
These statistics guide future applicants in understanding their odds and strategically tailoring their application process. They offer a window into the broader dynamics of college admissions.

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Most popular questions from this chapter

An oil company purchased an option on land in Alaska. Preliminary geologic studies assigned the following prior probabilities. \\[ \begin{aligned} P(\text { high-quality oil }) &=.50 \\ P(\text { medium-quality oil }) &=.20 \\ P(\text { no oil }) &=.30 \end{aligned} \\] a. What is the probability of finding oil? b. After 200 feet of drilling on the first well, a soil test is taken. The probabilities of finding the particular type of soil identified by the test follow. \\[ \begin{aligned} P(\text { soil } | \text { high-quality oil }) &=.20 \\ P(\text { soil } | \text { medium-quality oil }) &=.80 \\ P(\text { soil } | \text { no oil }) &=.20 \end{aligned} \\] How should the firm interpret the soil test? What are the revised probabilities, and what is the new probability of finding oil?

A decision maker subjectively assigned the following probabilities to the four outcomes of an experiment: \(P\left(E_{1}\right)=.10, P\left(E_{2}\right)=.15, P\left(E_{3}\right)=.40,\) and \(P\left(E_{4}\right)=.20 .\) Are these probability assignments valid? Explain.

The Powerball lottery is played twice each week in 28 states, the Virgin Islands, and the District of Columbia. To play Powerball a participant must purchase a ticket and then select five numbers from the digits 1 through 55 and a Powerball number from the digits 1 through 42\. To determine the winning numbers for each game, lottery officials draw 5 white balls out of a drum with 55 white balls, and 1 red ball out of a drum with 42 red balls. To win the jackpot, a participant's numbers must match the numbers on the 5 white balls in any order and the number on the red Powerball. Eight coworkers at the ConAgra Foods plant in Lincoln, Nebraska, claimed the record \(\$ 365\) million jackpot on February \(18,2006,\) by matching the numbers \(15-17-43-44-49\) and the Powerball number \(29 .\) A variety of other cash prizes are awarded each time the game is played. For instance, a prize of \(\$ 200,000\) is paid if the participant's five numbers match the numbers on the 5 white balls (Powerball website, March 19,2006 ). a. Compute the number of ways the first five numbers can be selected. b. What is the probability of winning a prize of \(\$ 200,000\) by matching the numbers on the 5 white balls? c. What is the probability of winning the Powerball jackpot?

The prior probabilities for events \(A_{1}, A_{2},\) and \(A_{3}\) are \(P\left(A_{1}\right)=.20, P\left(A_{2}\right)=.50,\) and \(P\left(A_{3}\right)=\) \(.30 .\) The conditional probabilities of event \(B\) given \(A_{1}, A_{2},\) and \(A_{3}\) are \(P\left(B | A_{1}\right)=.50\) \(P\left(B | A_{2}\right)=.40,\) and \(P\left(B | A_{3}\right)=.30\). a. Compute \(P\left(B \cap A_{1}\right), P\left(B \cap A_{2}\right),\) and \(P\left(B \cap A_{3}\right)\). b. Apply Bayes' theorem, equation \((4,19),\) to compute the posterior probability \(P\left(A_{2} | B\right)\). c. Use the tabular approach to applying Bayes' theorem to compute \(P\left(A_{1} | B\right), P\left(A_{2} | B\right)\), and \(P\left(A_{3} | B\right)\) .

The U.S. Census Bureau provides data on the number of young adults, ages \(18-24,\) who are living in their parents' home. \(^{1}\) Let \(M=\) the event a male young adult is living in his parents' home \(F=\) the event a female young adult is living in her parents' home If we randomly select a male young adult and a female young adult, the Census Bureau data enable us to conclude \(P(M)=.56\) and \(P(F)=.42\) (The World Almanac, 2006 ). The probability that both are living in their parents' home is .24 a. What is the probability at least one of the two young adults selected is living in his or her parents' home? b. What is the probability both young adults selected are living on their own (neither is living in their parents' home)?
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