91Ó°ÊÓ

A school has 250 employees categorized by task and gender in the following table. $$\begin{array}{|l|c|c|c|}\hline & \text { Teaching } & \text { Administrative } & \text { Support } \\\\\hline \text { Male } & 84 & 14 & 52 \\\\\hline \text { Female } & 56 & 26 & 18 \\\\\hline\end{array}$$ An employee is randomly selected. Let \(A\) be the event that he/she is an administrative staff member, T teaching staff, S support, \(M\) male, and F female. a) Write down the following probabilities: \(P(F), P(F \cap T), P\left(F \cup A^{\prime}\right), P\left(F^{\prime} | A\right)\) b) Which events are independent of \(F\), which are mutually exclusive to \(F\). Justify your choices. c) (i) Given that \(90 \%\) of teachers, as well as \(80 \%\) of the administrative staff and \(30 \%\) of the support staff, own cars, find the probability that a staff member chosen at random owns a car. (ii) Knowing that the randomly chosen staff member owns a car, find the probability that he/she is a teacher.

Step by step solution

01

Calculate Total and Female Probabilities

First, let's calculate the total number of employees and identify all females. From the table, we see there are 84 male teachers, 14 male administrators, 52 male support staff, 56 female teachers, 26 female administrators, and 18 female support staff.The total number of employees is: \[ \text{Total Employees} = 84 + 14 + 52 + 56 + 26 + 18 = 250 \]The total number of female employees is:\[ F = 56 + 26 + 18 = 100 \]Thus, the probability of choosing a female employee, \( P(F) \), is:\[ P(F) = \frac{100}{250} = 0.4 \]

02

Calculate Intersection and Complement Probabilities

The probability \( P(F \cap T) \) represents selecting a female teacher.The number of female teachers is:\[ F \cap T = 56 \]Therefore, the probability is:\[ P(F \cap T) = \frac{56}{250} = 0.224 \]To determine \( P(F \cup A') \), we need to find how many aren't administrative staff or are female. The count for female or non-administrative is:\[ M \cup A' = 250 - (14 + 26) = 210 \]Thus, \( P(F \cup A') = \frac{210}{250} = 0.84 \).

03

Calculating Conditional Probability

Next, calculate \( P(F' | A) \). First, we find \( A \) and \( F \cap A \):Number of administrative employees:\[ A = 14 + 26 = 40 \]Number of male (not female) among administrators:\[ F' \cap A = 14 \]Thus, the conditional probability is:\[ P(F' | A) = \frac{F' \cap A}{A} = \frac{14}{40} = 0.35 \]

04

Determine Independence and Exclusivity

Events \( F \) and \( A \) are independent if \( P(F \cap A) = P(F) \cdot P(A) \). Calculate \( P(A) \) = \( \frac{40}{250} = 0.16 \).\( P(F \cap A) = \frac{26}{250} = 0.104 \). Since \( 0.104 eq 0.4 \cdot 0.16 \), they are not independent.\( F \) and \( M \) are mutually exclusive since they cannot happen at the same time (\( P(F \cap M) = 0 \)).

05

Calculate Probability for Car Ownership

For car ownership, given probabilities are:\( P( ext{car | teaching}) = 0.9 \), \( P( ext{car | administrative}) = 0.8 \), \( P( ext{car | support}) = 0.3 \).Calculate total car ownership:\[ P( ext{car}) = (0.9 \cdot 140/250) + (0.8 \cdot 40/250) + (0.3 \cdot 70/250) \]\[ P( ext{car}) = 0.504 + 0.128 + 0.084 = 0.716 \]

06

Bayes' Theorem for Teacher Given Car Ownership

To find \( P(T | ext{car}) \), use Bayes’ Theorem:\[ P(T | ext{car}) = \frac{P( ext{car | T}) P(T)}{P( ext{car})} = \frac{0.9 \cdot \frac{140}{250}}{0.716} \]Compute:\[ P(T | ext{car}) = \frac{0.504}{0.716} \approx 0.704 \]

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.

Independent Events

Two events are considered independent if the occurrence of one event does not affect the probability of the other event occurring.
To determine if two events, say event A and event B, are independent, we use the formula:

• Event A and B being independent means that \( P(A \cap B) = P(A) \times P(B) \).

If this equation holds true, the events are independent; otherwise, they are not.
In the exercise, we checked whether being female (F) and being an administrative staff member (A) are independent events. We found that \( P(F \cap A) eq P(F) \times P(A) \), which indicates these events are not independent.
This is because the involvement of being female influences the probability of being in an administrative role, which demonstrates dependence.

Mutually Exclusive Events

Two events are mutually exclusive if they cannot occur simultaneously.
In simpler terms, when one event occurs, it excludes the possibility of the other occurring. The key characteristic is:

• If events A and B are mutually exclusive, then \( P(A \cap B) = 0 \).

In our specific problem, once we know someone is female (F), it confirms they cannot be male (M), and vice versa. Thus, the events female (F) and male (M) are mutually exclusive.
Understanding this concept helps differentiate situations where two events absolutely cannot happen together, as opposed to them simply being unlikely.
These insights are vital when categorizing types of probabilities and understanding event relationships.

Conditional Probability

Conditional probability deals with determining the probability that an event occurs, given that another event has already occurred.
It provides a way to focus on a particular subset of outcomes, reflective of additional given information. The formula for conditional probability is:

• \( P(A | B) = \frac{P(A \cap B)}{P(B)} \)

Given information changes our perspective on the likelihood of events.
In the context of this exercise, calculating \( P(F' | A) \) helped us understand the probability of a selected staff member not being female, given that they are administrative staff.
Such assessments are critical in scenarios where additional information influences event outcomes, providing more accurate probability estimates.

Bayes' Theorem

Bayes' Theorem provides a mathematical way to update our probability estimates based on new information, offering a dynamic framework for probability prediction.
It combines prior knowledge with new evidence, using the formula:

• \( P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)} \)

This theorem is particularly useful when we're trying to find the probability of an event after obtaining more information.
In our exercise, we employed Bayes' Theorem to find the probability that a randomly chosen staff member is a teacher, given that this person owns a car.
This application helps in effectively revising probability estimates when new insights emerge, making Bayes' Theorem a powerful statistical tool for decision-making and predictive analytics.