91Ó°ÊÓ

A company uses three different assembly lines \(A_{1}, A_{2}\), and \(A_{3}\) -to manufacture a particular component. Of those manufactured by \(A_{1}, 5 \%\) need rework to remedy a defect, whereas \(8 \%\) of \(A_{2}\) 's components and \(10 \%\) of \(A_{3}\) 's components need rework. Suppose that \(50 \%\) of all components are produced by \(A_{1}\), whereas \(30 \%\) are produced by \(A_{2}\) and \(20 \%\) come from \(A_{3}\). a. Construct a tree diagram with first-generation branches corresponding to the three lines. Leading from each branch, draw one branch for rework (R) and another for no rework (N). Then enter appropriate probabilities on the branches. b. What is the probability that a randomly selected component came from \(A_{1}\) and needed rework? c. What is the probability that a randomly selected component needed rework?

Step by step solution

01

Construct a Tree Diagram

Start by drawing three branches from a single point representing the three assembly lines \(A_{1}, A_{2}\), and \(A_{3}\). From each branch, draw two more branches representing the possibilities of rework 'R' (with defect) and 'N' (without defect). Label the branches with their corresponding probabilities. The first set of branches is the probability of each assembly line being used (i.e., \(0.50\) for \(A_{1}\), \(0.30\) for \(A_{2}\), and \(0.20\) for \(A_{3}\)). The following set of branches represent the work coming from these assembly lines, which includes 'R' for rework needed (i.e., \(0.05\) for \(A_{1}\), \(0.08\) for \(A_{2}\), and \(0.10\) for \(A_{3}\)) and 'N' for no rework needed.

02

Compute the Joint Probability for Assembly Line \(A_{1}\) and Rework Needed

This is computed by multiplying the probability of picking \(A_{1}\) by the probability of a component from \(A_{1}\) needing rework. Mathematically, it would be \(P(A_{1} and R) = P(A_{1}) * P(R|A_{1}) = 0.50 * 0.05 = 0.025\).

03

Calculate the Total Probability of Needed Rework

The total probability of a component needing rework is the sum of the joint probabilities of needing rework from each assembly line. So, \(P(R) = P(A_{1} and R) + P(A_{2} and R) + P(A_{3} and R) = (P(A_{1}) * P(R|A_{1})) + (P(A_{2}) * P(R|A_{2})) + (P(A_{3}) * P(R|A_{3})) = (0.50*0.05) + (0.30*0.08) + (0.20*0.10) = 0.025 + 0.024 + 0.020 = 0.069\).

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Key Concepts

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

Conditional Probability and Its Role

Conditional probability is a key concept in probability theory that is used to find the likelihood of an event, given that another event has taken place. In the context of the assembly line problem, the probability that a component needs rework given it came from a specific assembly line is an example of conditional probability. For assembly line analysis:

• The conditional probability of needing rework if the component comes from assembly line \(A_{1}\) is \(0.05\).
• If it's from \(A_{2}\), the probability is \(0.08\).
• From \(A_{3}\), it's \(0.10\).

The formula for conditional probability is \(P(R|A_i)\), where \(R\) is the rework event and \(A_i\) is the event that the component comes from the \(i\)-th assembly line. Understanding conditional probability helps to predict outcomes more accurately based on specific conditions.

Understanding Probability Theory Basics

Probability theory forms the foundation of analyzing scenarios like assembly line operations. In this scenario:

• We start by considering the probability of choosing any assembly line, which are \(0.50\) for \(A_{1}\), \(0.30\) for \(A_{2}\), and \(0.20\) for \(A_{3}\).
• Then, each line has its own probability of producing defective components.
• The principle of independence implies these probabilities are separate; hence, we can multiply them to find joint probabilities.

For example, the joint probability that a component is from \(A_{1}\) and needs rework is calculated as \(0.025\). Using probability theory, you can predict and calculate events in complex systems.

Assembly Line Analysis Techniques via Tree Diagrams

In assembly line analysis, tree diagrams are a visual tool that can help simplify complex probability scenarios. They break down each decision path a component might follow and attach probabilities. This allows us to see the flow of events and their likelihoods clearly.For the assembly line exercise:

• Start by drawing a main branch for each assembly line \(A_{1}, A_{2}, A_{3}\).
• From each of these branches, draw additional branches for 'Rework' and 'No Rework'.
• Label each branch with its probability; for example, \(0.05\) rework probability for \(A_{1}\).

Using a tree diagram can simplify the process of finding probabilities of complex multi-step processes such as those found in industrial operations. They provide a structured way to consider all possible outcomes, helping to ensure accurate probability calculations and better decision-making.