Problem 35 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The admissions office of a private university released the following admission data for the preceding academic year: From a pool of 3900 male applicants, \(40 \%\) were accepted by the university and \(40 \%\) of these subsequently enrolled. Additionally, from a pool of 3600 female applicants, \(45 \%\) were accepted by the university and \(40 \%\) of these subsequently enrolled. What is the probability that a. A male applicant will be accepted by and subsequently will enroll in the university? b. A student who applies for admissions will be accepted by the university? c. A student who applies for admission will be accepted by the university and subsequently will enroll?

Step-by-Step Solution

Answer

a. The probability that a male applicant will be accepted by and subsequently will enroll in the university is \(\frac{624}{7500}\). b. The probability that a student who applies for admissions will be accepted by the university is \(P(A) = 0.40 * \frac{3900}{7500} + 0.45 * \frac{3600}{7500}\). c. The probability that a student who applies for admission will be accepted by the university and subsequently will enroll is \(P(E \cap A) = \frac{624}{7500} + (0.40 * (0.45 * \frac{3600}{7500}))\).

Step 1: First, let's denote some events to make our calculations easier. Let \(A\) be the event that a student is accepted. Let \(E\) be the event that a student subsequently enrolls. Let \(M\) be the event that the applicant is male, and \(F\) be the event that the applicant is female. We are given the following probabilities: - The probability of acceptance given that the applicant is male: \(P(A|M) = 0.40\) - The probability of enrollment given that the applicant is male and accepted: \(P(E|A \cap M) = 0.40\) - The probability of acceptance given that the applicant is female: \(P(A|F) = 0.45\) - The probability of enrollment given that the applicant is female and accepted: \(P(E|A \cap F) = 0.40\) #Step 2: Find the probability that a male applicant will be accepted and subsequently enroll in the university (P(E ∩ A ∩ M))#

We want to find the probability of a male applicant being accepted and subsequently enrolling in the university, i.e., \(P(E \cap A \cap M)\). Using conditional probability formula, we can rewrite this as: \(P(E \cap A \cap M) = P(E| A \cap M) * P(A \cap M)\) We already know the probability of enrollment given that the applicant is male and accepted: \(P(E|A \cap M) = 0.40\) Now, we need to find the probability of acceptance for a male applicant (simply \(P(A \cap M)\)). We'll use the marginal probability formula: \(P(A \cap M) = P(A|M) * P(M)\) We know \(P(A|M) = 0.4\) and we can calculate the probability of an applicant being male as: \[P(M) = \frac{\text{Number of Male Applicants}}{\text{Total Number of Applicants}} = \frac{3900}{ (3900+3600)} = \frac{3900}{7500} \] Now, we can calculate \(P(A \cap M)\): \[P(A \cap M) = P(A|M) * P(M) = 0.40 * \frac{3900}{7500} \] Finally, we can calculate \(P(E \cap A \cap M)\): \[P(E \cap A \cap M)= P(E| A \cap M) * P(A \cap M) = 0.40 * (0.40 * \frac{3900}{7500}) = \frac{624}{7500} \] a. The probability that a male applicant will be accepted by and subsequently will enroll in the university is \(\frac{624}{7500}\). #Step 3: Find the probability that a student who applies for admissions will be accepted by the university (P(A))#

Step 2: To find the probability that a student who applies for admissions will be accepted by the university, we need to calculate the marginal probability of acceptance: \(P(A)\). We can use the law of total probability to obtain this: \[P(A) = P(A|M) * P(M) + P(A|F) * P(F)\] We already calculated \(P(A|M) * P(M)\) and know \(P(A|F) = 0.45\). Now, we just need to find the probability of an applicant being female: \[P(F)=\frac{\text{Number of Female Applicants}}{\text{Total Number of Applicants}} = \frac{3600}{ (3900+3600)} = \frac{3600}{7500} \] Now, we can calculate \(P(A)\): \[P(A) = P(A|M) * P(M) + P(A|F) * P(F) = 0.40 * \frac{3900}{7500} + 0.45 * \frac{3600}{7500}\] b. The probability that a student who applies for admissions will be accepted by the university is \(P(A) = 0.40 * \frac{3900}{7500} + 0.45 * \frac{3600}{7500}\). #Step 4: Find the probability that a student who applies for admission will be accepted by the university and subsequently enroll (P(E ∩ A))#

Lastly, to find the probability that a student who applies for admission will be accepted by the university and subsequently enroll, we need to calculate \(P(E \cap A)\). Similar to step 3, we can use the law of total probability: \[P(E \cap A) = P(E \cap A \cap M) + P(E \cap A \cap F)\] We've already calculated the probability for male students (\(P(E \cap A \cap M)\)) in step 2. Now, we just need to find the probability of acceptance and enrollment for female students (\(P(E \cap A \cap F)\)): Using conditional probability formula: \(P(E \cap A \cap F) = P(E| A \cap F) * P(A \cap F)\) We know \(P(E| A \cap F) = 0.40\), and we can find the probability of acceptance for a female applicant (simply \(P(A \cap F)\)): \[P(A \cap F) = P(A|F) * P(F) = 0.45 * \frac{3600}{7500} \] Then we can calculate \(P(E \cap A \cap F)\): \[P(E \cap A \cap F)= P(E| A \cap F) * P(A \cap F) = 0.40 * (0.45 * \frac{3600}{7500})\] Now, we can calculate the probability that a student who applies for admission will be accepted by the university and subsequently enroll: \[P(E \cap A) = P(E \cap A \cap M) + P(E \cap A \cap F) = \frac{624}{7500} + (0.40 * (0.45 * \frac{3600}{7500}))\] c. The probability that a student who applies for admission will be accepted by the university and subsequently will enroll is \(P(E \cap A) = \frac{624}{7500} + (0.40 * (0.45 * \frac{3600}{7500}))\).

Key Concepts

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. This concept is essential in understanding various real-world processes, such as admission processes in universities. For example, when evaluating university applications, we might want to know the probability of a student enrolling given that they have been accepted. Using conditional probability, we identify how two events relate to each other, such as acceptance and subsequent enrollment. In mathematical terms, if we are given two events, A and B, the conditional probability of A given B is expressed using the formula: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]This means that the probability of A happening, when B is known, depends on the joint probability of both A and B happening, divided by the probability of B alone. In our university case, we determined that the probability of enrollment given acceptance (for males and females), as part of the larger admission statistics, helps policy makers understand applicant behavior more profoundly. By diving into conditional probabilities, universities and institutions can strategize better for future intakes by knowing past applicant behaviors and outcomes.

Total Probability

The law of total probability provides a way to calculate the probability of an event based on several different scenarios that could lead to that event. It offers a comprehensive view and is especially helpful in complex situations like admission processes where multiple paths (e.g., male or female applicants) can lead to the same outcome. In the context of university admissions, applying this law helps compute the probability of a student getting accepted without distinguishing between male or female applicants.Using the law of total probability, to find the probability of acceptance by the university, we consider both male and female applicant pools. The formula for total probability comes into play as:\[ P(A) = P(A|M) \cdot P(M) + P(A|F) \cdot P(F) \]Here, the overall probability of acceptance (P(A)) comes from the probability of acceptance among males and females, weighted by their respective frequencies in the total applicant pool.This method of combining individual probabilities provides a balanced perspective across different applicant groups. Hence, it enables institutions to make data-driven admissions policies and foresee acceptance patterns within diverse populations.

Gender-Based Statistics

Understanding gender-based statistics can significantly illuminate trends in university admissions and other processes. By evaluating data separately for males and females, we capture potential discrepancies that could exist between different groups. In the given admissions scenario, gender-based statistics reveal different acceptance rates for male and female applicants. For example, the data showed that 40% of male and 45% of female applicants were accepted. Collecting this gender-segregated data helps universities discern whether their acceptance policies are resulting in fair outcomes for each gender. It also uncovers potential areas for improvement, fostering equality and diversity on campuses. With gender-based insights, schools can:

• Tailor outreach and support programs specifically targeting underrepresented groups.
• Analyze the impact of their gender-related initiatives over time.
• Ensure transparency and fairness in their admission processes.

In essence, acknowledging gender-based statistics is not just about maintaining fairness but also about enhancing the overall educational environment by recognizing the diversity of applicant backgrounds.

Acceptance Rates

Acceptance rates provide a snapshot of how selective a university or program is during the admissions process. These rates represent the percentage of applicants who are offered admission out of the total number of candidates. By calculating and analyzing acceptance rates, institutions can gauge their desirability and competitiveness. In our example, the acceptance rates are calculated separately for male and female applicants, which in turn influences the overall acceptance rate. If a university receives 3900 male applications and accepts 40% of them, this directly affects the perceived selectiveness of the institution. It's crucial to present acceptance rates in such a way that reflects the true nature of the applicant pool, considering all contributing factors such as gender, background, and more. Acceptance rates can:

• Help prospective students understand their chances of gaining admission.
• Guide universities in modifying strategies to meet target enrollment numbers.
• Reflect trends in applicant preferences over different academic cycles.

Overall, leveraging acceptance rates wisely ensures a balanced approach to admissions that can sustain the institution's reputation and meet long-term strategic goals.