91Ó°ÊÓ

Screening job applicants. A company retains a psychologist to assess whether job applicants are sulted for assembly-line work. The psychologist classifies applicants as one of \(A\) (well suited), \(B\) (marginal), or \(C\) (not suited). The company is concerned about the event \(D\) that an employee leaves the company within a year of being hired. Data on all people hired in the past five years give these probabilities: \(P(A)=0.4 P(B)=0.3 P(C)=0.3 P(A\) and \(D)=0.1 P(B\) and \(D)=0.1 P(C\) and \(D)=0.2\) \(P(A)=0 . A \quad P(B)=0.3 \quad P(C)=0.3\) \(P(A\) and \(D)=0.1 \quad P(B\) and \(D)=0.1 \quad P(C\) and \(D)=0.2\) Sketch a Venn diagram of the events \(A, B, C\), and \(D\) and mark on your diagram the probabilities of all combinations of psychological assessment and leaving (or not) within a year. What is \(P(D)\), the probability that an employee leaves within a year?

Step by step solution

01

Understand the Problem

We have three types of applicants classified by a psychologist as well-suited ( A ), marginal ( B ), or not suited ( C ). We are also concerned with the event ( D ) where an employee leaves within a year. We need to use the given probabilities to find the overall probability of ( D ).

02

Calculate Total Probability of D

The total probability of an employee leaving within a year, (P(D)), can be calculated using the law of total probability. This involves adding up the probabilities of (D) occurring in conjunction with each category. \[P(D) = P(A \text{ and } D) + P(B \text{ and } D) + P(C \text{ and } D)\] Substituting the given values: \[P(D) = 0.1 + 0.1 + 0.2 = 0.4\]

03

Sketch Venn Diagram

Draw a Venn diagram with three circles for events A , B , and C . Inside each circle, note the probabilities of leaving ( D ): 0.1 for A and D , 0.1 for B and D , and 0.2 for C and D . The remaining probability in each circle represents not leaving within a year.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Venn Diagram

A Venn diagram is an excellent tool for visualizing probabilities involving multiple events. It helps us understand how different sets interact with each other. In the context of job applicants, we consider three sets:

• \(A\): Applicants classified as well suited
• \(B\): Applicants classified as marginal
• \(C\): Applicants classified as not suited

We also have an event \(D\), where an employee leaves within a year of being hired. To sketch a Venn diagram for this exercise, imagine three overlapping circles representing \(A\), \(B\), and \(C\). Then you would mark within these circles the probability of \(D\) occurring in each assessment category:

• For \(A\) and \(D\): Probability is 0.1
• For \(B\) and \(D\): Probability is 0.1
• For \(C\) and \(D\): Probability is 0.2

The visual representation helps confirm the overlapping and non-overlapping parts of these events, making it easier to perform quick checks or adjustments.

Law of Total Probability

The law of total probability helps us find the overall probability of an event by considering all potential pathways to its occurrence. For our situation, we wish to find the total probability that an employee leaves within a year, denoted as \(P(D)\).
To apply the law, we sum the probabilities of \(D\) given each applicant category:

• \(P(A \text{ and } D)\)
• \(P(B \text{ and } D)\)
• \(P(C \text{ and } D)\)

The equation is:\[P(D) = P(A \text{ and } D) + P(B \text{ and } D) + P(C \text{ and } D)\]Substituting the given values yields:\[P(D) = 0.1 + 0.1 + 0.2 = 0.4\]This tells us that the overall likelihood of an employee leaving within a year is 40%. This law simplifies the complexity of dealing individually with multiple probabilities, offering a structured approach to encompassing all scenarios.

Statistical Analysis

Statistical analysis is crucial for interpreting data and drawing meaningful insights. In this job applicant screening context, it involves using probabilities calculated from historical hiring data to forecast future trends.
The probabilities for each applicant category help evaluate the effectiveness of the psychological assessment. By knowing, for instance, that 40% of all new hires leave within a year, the company can consider whether its hiring strategy is effective or if there may be improvement areas.
Analysis is typically done by comparing these calculated probabilities to industry averages or internal benchmarks. The insights can help companies refine their hiring processes, target better fitting candidates, and ultimately reduce turnover rates.

Job Applicant Screening

In job applicant screening, companies employ various methods to assess potential employees, ensuring they fit the role's requirements. Screening often involves psychological assessments, which help categorize applicants based on their suitability for the job.
In this exercise, the psychologist classifies applicants into three categories:

• \(A\): Well suited
• \(B\): Marginal
• \(C\): Not suited

These classifications aim to predict job success and longevity. The company is particularly interested in reducing employee turnover, measured by those leaving within a year (the event \(D\)).
Effective screening can optimize recruitment, reducing costs related to re-hiring and training new employees. By analyzing the probabilities of turnover among each classification, companies can fine-tune their hiring criteria, ensuring the selection of candidates aligned with their long-term employment goals.