91Ó°ÊÓ

A construction firm is currently working on three different buildings. Let \(A_{i}\) denote the event that the \(i\) th building is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in terms of \(A_{1}, A_{2}\), and \(A_{3}\), draw a Venn diagram, and shade the region corresponding to each one. a. At least one building is completed by the contract date. b. All buildings are completed by the contract date. c. Only the first building is completed by the contract date. d. Exactly one building is completed by the contract date. e. Either the first building or both of the other two buildings are completed by the contract date.

Step by step solution

01

Define Event A

Event A represents the completion of the buildings. Let \(A_1\) denote the event that building 1 is completed, \(A_2\) denote the completion of building 2, and \(A_3\) denote the completion of building 3 by the contract date.

02

Part a: At Least One Building Completed

The event that at least one building is completed is represented by the union of the events \(A_1\), \(A_2\), or \(A_3\). This can be expressed as \(A_1 \cup A_2 \cup A_3\). In a Venn diagram, shade any portion that includes at least one of the events \(A_1\), \(A_2\), or \(A_3\).

03

Part b: All Buildings Completed

The event that all buildings are completed is the intersection of all three events \(A_1\), \(A_2\), and \(A_3\). This can be represented by \(A_1 \cap A_2 \cap A_3\). Shade only the central portion where all three circles overlap in the Venn diagram.

04

Part c: Only the First Building Completed

This occurs when event \(A_1\) happens, but events \(A_2\) and \(A_3\) do not. Mathematically, this can be described as \(A_1 \cap A_2^c \cap A_3^c\). In the Venn diagram, shade only the part of \(A_1\) that is outside \(A_2\) and \(A_3\).

05

Part d: Exactly One Building Completed

This event requires either \(A_1\), \(A_2\), or \(A_3\) to occur, but not in any combination. It is represented as \((A_1 \cap A_2^c \cap A_3^c) \cup (A_1^c \cap A_2 \cap A_3^c) \cup (A_1^c \cap A_2^c \cap A_3)\). In the Venn diagram, shade exclusively the non-overlapping parts of each individual circle.

06

Part e: Either First Building or Other Two Completed

This situation is satisfied if \(A_1\) is completed or both \(A_2\) and \(A_3\) are completed. It can be represented as \(A_1 \cup (A_2 \cap A_3)\). Shade the entire section of \(A_1\) and the overlapping part of \(A_2\) and \(A_3\) in the Venn diagram.

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Key Concepts

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

Set Operations

In probability theory, set operations allow us to handle and analyze different events systematically. Set operations such as union, intersection, and complementation are essential for understanding how different events relate to one another.

• Union: The union of two or more sets involves combining all the elements that are in either set. In probability, the union of events represents any of the events happening. It is denoted by the symbol \( \cup \), and for events \( A_1, A_2, \) and \( A_3 \) it would be written as \( A_1 \cup A_2 \cup A_3 \).
• Intersection: The intersection of sets includes only the elements that are present in all of the sets. In probability, this translates to an event where all given events occur simultaneously. It is represented by the symbol \( \cap \). For example, \( A_1 \cap A_2 \cap A_3 \) means all three events occur at the same time.
• Complementation: The complement of a set includes all the elements not in the set. Complementation is used to express events not happening. This is denoted by the superscript \( c \), such as \( A_1^c \), meaning event \( A_1 \) does not occur.

Events in Probability

Events in probability represent the basic outcomes of a probabilistic experiment. An event can be simple, involving a single outcome, or complex, encompassing multiple outcomes. Understanding these events is crucial to solving probability problems.

Let's explore this through the exercise related to the construction firm:

• At least one building completed: This is the union of events \( A_1, A_2, \) or \( A_3 \), reflecting the completion of at least one building. It may occur if any or all of the individual events happen.
• All buildings completed: An intersection of \( A_1 \cap A_2 \cap A_3 \) indicates all buildings are completed, as this event requires that all three occur together.
• Only the first building completed: Here, event \( A_1 \cap A_2^c \cap A_3^c \) signifies only the first building is completed while the others remain unfinished.
• Exactly one building completed: This involves the union of specific intersections, such as \((A_1 \cap A_2^c \cap A_3^c)\), representing situations where only one event occurs, excluding any overlap.
• Either the first building or others: This complex event \( A_1 \cup (A_2 \cap A_3) \), shows either the first building or both of the others are completed, combining union and intersection ideas.

Venn Diagrams

Venn diagrams are a visual tool used to represent set operations and events. They help illustrate how different sets overlap and relate in terms of union, intersection, and complementation.

To apply Venn diagrams to probability, consider events \( A_1, A_2, \) and \( A_3 \) representing the completion of buildings one, two, and three:

• Union: Shaded areas covering any part of the circles show \( A_1 \cup A_2 \cup A_3 \), meaning any of the buildings can complete.
• Intersection: The central part where all circles intersect, such as \( A_1 \cap A_2 \cap A_3 \), illustrates all buildings completed.
• Complementation: Involves shading the outer areas of respective circles for \( A_1^c \), excluding sections displaying when \( A_1 \) happens.

Using Venn diagrams can simplify understanding and solving complex probability problems by providing a clearer picture of how events interrelate. They highlight the relationships visually, helping convey intricate details with simplicity.