Question

Consider the chance experiment in which an automobile is selected and both the number of defective headlights \((0,1\), or 2\()\) and the number of defective tires \((0,1\), 2,3, or 4 ) are determined. a. Display possible outcomes using a tree diagram. b. Let \(A\) be the event that at most one headlight is defective and \(B\) be the event that at most one tire is defective. What outcomes are in \(A^{C} ?\) in \(A \cup B ?\) in \(A \cap B ?\) c. Let \(C\) denote the event that all four tires are defective. Are \(A\) and \(C\) disjoint events? Are \(B\) and \(C\) disjoint?

Short answer

The outcomes in Ac are those with 2 defective headlights, regardless of the number of defective tires. Outcomes in A∪B are those with up to 1 defective tire, up to 1 defective headlight, or both. The outcomes in A∩B are those with up to 1 defective tire and 1 defective headlight at the same time. A and C are not disjoint events while B and C are disjoint events.

Step-by-step solution

Constructing the Tree Diagram

In a tree diagram, we can display all possible outcomes. For the first branch, we choose either 0, 1, or 2 defective headlights. Then, for each of these branches, we add another branch for the tires, choosing between 0, 1, 2, 3, or 4 defective tires.

Determine Outcomes in Ac

The event A denotes the situation when at most one headlight is defective. Therefore, the complement of event A, denoted by Ac, includes the outcomes when more than one (2 in this case) headlight is defective. Regardless of the state of the tires, any outcome with 2 defective headlights is in Ac.

Determine Outcomes in A∪B

The union of two events A and B, denoted by A∪B, includes all outcomes that are in either event A, event B, or both. Event A includes outcomes with at most one defective headlight and event B includes outcomes with at most one defective tire. So any outcomes with up to 1 defective component (either tire or headlight) are included in A∪B.

Determine Outcomes in A∩B

The intersection of two events A and B, denoted by A∩B, includes all outcomes that are in both event A and event B. Therefore, all outcomes with at most 1 defective headlight and at most 1 defective tire are included in A∩B.

Determine if A and C are disjoint

Two events are disjoint if they have no outcomes in common. Event A includes outcomes with at most 1 defective headlight, while event C includes outcomes with all four tires defective. Since an outcome with more than one defective headlight can also have all tires defective, A and C are not disjoint.

Determine if B and C are Disjoint

Event B includes outcomes with at most 1 defective tire, while event C includes outcomes with all 4 tires defective. These two events cannot both occur at the same time, meaning B and C are disjoint.

Key concepts

Defective Automobile Components

When analyzing the quality of automotive production, identifying and understanding the occurrence of defective automobile components is critical. In the given exercise, the focus is on two key parts of a vehicle: headlights and tires. A fundamental approach to analyzing such defects is via probability.

For instance, let's consider the tree diagram statistics aspect of the problem: the tree diagram serves as a visual tool that systematically displays all possible outcomes of defective components in headlights and tires. Each branch represents a potential outcome, illustrating the logical progression of events. The depth of the diagram will depend on the number of components considered—in this case, headlights and tires, with several levels of defectiveness (0, 1, 2 for headlights, and 0 to 4 for tires).

By using this method, manufacturers and quality control teams can predict and analyze the likelihood of defects, which assists in the improvement of manufacturing processes and product quality. Understanding these probabilities helps in resource allocation for repairs and replacements, and in some cases, can lead to proactive measures to reduce the likelihood of defects occurring in the first place.

Probability Events

Understanding Events and Their Complements

Probability events are the foundational blocks in the study of probability and statistics. They are the outcomes or sets of outcomes that we focus on when conducting a probability experiment. In the context of the textbook exercise, we are asked to consider events such as having at most one defective headlight or tire.

In practical terms, an event might represent a scenario where a car passes a quality inspection (event A), or perhaps where it fails due to tire issues (event B). The complement of an event (such as Ac) represents all the outcomes not included in the original event, which here would be the cars that fail the inspection due to headlight issues. The union of events (A∪B) considers either of the issues being present, while the intersection (A∩B) requires both to be simultaneously true. Vehicle manufactures can benefit from understanding these probabilities to manage quality and predict potential recalls or warranty claims.

Disjoint events, as also inquired in the exercise, are those that cannot occur simultaneously. For example, a production line cannot produce a car that simultaneously passes and fails the quality inspection on headlights. This concept helps industries in schematic risk management and in devising strategies that mitigate multiple points of failure.

Set Theory in Statistics

Connecting Set Theory to Probabilistic Outcomes

Set theory is an elegant and powerful tool in statistics that deals with the study of collections, or 'sets', of objects. We use set theory to understand and formalize the relationships between different probabilistic events. In the problem we're discussing, sets and their properties illuminate the outcomes of defective automotive parts.

Included within set theory is the idea of unions, intersections, and complements. The operation of taking a union or intersection essentially reflects combining or overlapping sets, respectively. When we look at the union (A∪B), we are focusing on all the outcomes included in either set A or set B, or in both. In contrast, intersections (A∩B) involve situations that are true for both sets. Finally, a complement of a set (Ac) represents all elements not in set A.

These concepts empower statisticians and quality control managers to predict the frequency of defects and deal with them effectively. By categorizing possible outcomes into sets, they can use mathematical rigor to support decision-making processes in production, quality assurance, and customer service operations.