/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 32 Quaurr CoNrroL. An automobile ma... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 32

Quaurr CoNrroL. An automobile manufacturer obtains the microprocessors used to regulate fuel consumption in its automobiles from three microelectronic firms: \(\mathrm{A}, \mathrm{B}\), and C. The quality-control department of the company has determined that \(1 \%\) of the microprocessors produced by firm \(A\) are defective, \(2 \%\) of those produced by firm \(B\) are defective, and \(1.5 \%\) of those produced by firm \(\mathrm{C}\) are defective. Firms \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\) supply \(45 \%, 25 \%\), and \(30 \%\), respectively, of the microprocessors used by the company. What is the probability that a randomly selected automobile manufactured by the company will have a defective microprocessor?

Short Answer

Expert verified
The probability that a randomly selected automobile manufactured by the company will have a defective microprocessor is 1.4%.

Step by step solution

01

Understand the law of total probability

The law of total probability states that the probability of an event \(A\) can be expressed as a weighted sum of the conditional probabilities of A given \(B_1, ..., B_n\), where \(B_1,...,B_n\) are mutually exclusive and exhaustive events. In our case, events \(B_i\) are the different firms (\(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\)) that produce the microprocessors.
02

Define the events

Let \(D\) be the event of selecting a defective microprocessor, and let \(A, B\), and \(C\) be the events of selecting a microprocessor from each respective firm.
03

Write out the probabilities

We are given the following probabilities: \(P(D|A)=0.01, P(D|B)=0.02, P(D|C)=0.015, P(A)=0.45, P(B)=0.25,\) and \(P(C)=0.3\).
04

Apply the law of total probability

Using the law of total probability, we can find the probability of a randomly selected automobile having a defective microprocessor: \(P(D) = P(D|A)P(A) + P(D|B)P(B) + P(D|C)P(C)\).
05

Compute the probability of defective microprocessor

Plug in the given probabilities into the equation: \(P(D) = (0.01)(0.45) + (0.02)(0.25) + (0.015)(0.3)\). Now, perform the arithmetic: \(P(D) = 0.0045 + 0.005 + 0.0045\). Finally, add the probabilities to get the overall probability of a defective microprocessor: \(P(D) = 0.014\). Thus, the probability that a randomly selected automobile manufactured by the company will have a defective microprocessor is 1.4%.

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 91影视!

Key Concepts

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

Conditional Probability
Understanding conditional probability is vital for solving problems in probability theory that involve certain events occurring given the presence of other events.

Conditional probability is defined as the probability of one event, A, happening given that another event, B, has already occurred, denoted as P(A|B). In simpler terms, it reflects how the likelihood of event A changes when we have information about event B.

In the exercise, we calculated the probabilities of a microprocessor being defective (D) given it's produced by a specific firm. This means we looked at the probability of defects (P(D|A), P(D|B), P(D|C)) conditioned on the firm that produced the microprocessor (A, B, or C).
Probability Theory
Probability theory is the mathematical framework for quantifying uncertainty and making predictions about the likelihood of various outcomes.

It is built on a set of axioms that govern how probabilities are assigned and manipulated, and it leverages several fundamental rules like the rule of multiplication for independent events and the law of total probability. The exercise demonstrated the application of the law of total probability, which allows us to combine conditional probabilities to find the total probability of an event.
Quality Control Statistics
Quality control statistics use statistical methods to monitor and manage the quality of production.

By employing techniques like hypothesis testing, control charts, and probability calculations, quality control ensures that products meet certain standards and that the process variability is kept to a minimum.

Application in Exercise

Our exercise centered around defective microprocessors, showcasing how probabilities can inform quality control decisions. For instance, understanding the defect rates from different suppliers helps a manufacturer to manage risks and potentially revise supplier choices or quality processes.
Finite Mathematics
Finite mathematics refers to areas of mathematics that deal with finite, discrete quantities.

It includes topics such as logic, set theory, combinatorics, graph theory, and statistics. Probability, a significant part of finite mathematics, has practical applications in various fields, including computer science, economics, and operations research. The problem from the exercise uses finite mathematics to analyze discrete events (defective or non-defective microprocessors) and calculate the overall probability of an event occurring.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In an online survey of 1962 executives from 64 countries conducted by Korn/Ferry International between August and October 2006, the executives were asked if they would try to influence their children's career choices. Their replies: A (to a very great extent), B (to a great extent), \(\mathrm{C}\) (to some extent), D (to a small extent), and \(\mathrm{E}\) (not at all) are recorded below: $$\begin{array}{lccccc} \hline \text { Answer } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Respondents } & 135 & 404 & 1057 & 211 & 155 \\ \hline \end{array}$$ What is the probability that a randomly selected respondent's answer was \(\mathrm{D}\) (to a small extent) or \(\mathrm{E}\) (not at all)?

In "The Numbers Game," a state lottery, four numbers are drawn with replacement from an urn containing balls numbered \(0-9\), inclusive. Find the probability that a ticket holder has the indicated winning ticket. One digit (the first, second, third, or fourth digit)

A survey involving 400 likely Democratic voters and 300 likely Republican voters asked the question: Do you support or oppose legislation that would require trigger locks on guns, to prevent misuse by children? The following results were obtained: $$\begin{array}{lcc} \hline \text { Answer } & \text { Democrats, \% } & \text { Republicans, \% } \\\ \hline \text { Support } & 88 & 71 \\ \hline \text { Oppose } & 7 & 20 \\ \hline \text { Don't know/refused } & 5 & 9 \\ \hline \end{array}$$ If a randomly chosen respondent in the survey answered "support," what is the probability that he or she is a likely Republican voter?

Two hundred workers were asked: Would a better economy lead you to switch jobs? The results of the survey follow: $$ \begin{array}{lccccc} \hline \text { Answer } & \begin{array}{c} \text { Very } \\ \text { likely } \end{array} & \begin{array}{c} \text { Somewhat } \\ \text { likely } \end{array} & \begin{array}{c} \text { Somewhat } \\ \text { unlikely } \end{array} & \begin{array}{c} \text { Very } \\ \text { unlikely } \end{array} & \begin{array}{c} \text { Don't } \\ \text { know } \end{array} \\ \hline \text { Respondents } & 40 & 28 & 26 & 104 & 2 \\ \hline \end{array} $$ If a worker is chosen at random, what is the probability that he or she a. Is very unlikely to switch jobs? b. Is somewhat likely or very likely to switch jobs?

At a certain medical school, \(\frac{1}{7}\) of the students are from a minority group. Of those students who belong to a minority group, \(\frac{1}{3}\) are black. a. What is the probability that a student selected at random from this medical school is black? b. What is the probability that a student selected at random from this medical school is black if it is known that the student is a member of a minority group?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.