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An engineering construction firm is currently working on power plants at three different sites. Let \(A_{i}\) denote the event that the plant at site \(i\) 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 plant is completed by the contract date. b. All plants are completed by the contract date. c. Only the plant at site 1 is completed by the contract date. d. Exactly one plant is completed by the contract date. e. Either the plant at site 1 or both of the other two plants are completed by the contract date.

Step by step solution

01

Understanding the Required Concepts

We are given three events: \( A_1 \), \( A_2 \), \( A_3 \) which represent the completion of plants at sites 1, 2, and 3, respectively, by their due date. These events can be combined using the union, intersection, and complementation operations to describe the scenarios posed in the problem. Let's understand how to use these operations to describe each scenario.

02

At least one plant is completed

To express the event that at least one plant is completed by its contract date, we use the union of all three events. This is because the union of these events represents completing at least one of them. More formally, it is described by the union:\[ A_1 \cup A_2 \cup A_3 \]This set includes any scenario where at least one of the plants is completed on time.

03

All plants are completed

For all plants to be completed by the contract date, each of the events must occur simultaneously. This is represented by the intersection of the three sets:\[ A_1 \cap A_2 \cap A_3 \]The intersection ensures that all three events occur, meaning all plants are completed on time.

04

Only the plant at site 1 is completed

Only the plant at site 1 being completed means \( A_1 \) occurs, but \( A_2 \) and \( A_3 \) do not. Therefore, we represent this by:\[ A_1 \cap A_2^c \cap A_3^c \]This expression specifies that site 1 is completed, while sites 2 and 3 are not.

05

Exactly one plant is completed

Exactly one plant completed means one of the events occurs, and the other two do not. This is captured by:\[(A_1 \cap A_2^c \cap A_3^c) \cup (A_1^c \cap A_2 \cap A_3^c) \cup (A_1^c \cap A_2^c \cap A_3)\]Each term in the union represents a different single plant being completed on time.

06

Either plant at site 1 or both others completed

This condition is satisfied if either the plant at site 1 is completed or both other plants (sites 2 and 3) are completed. The expression is:\[ A_1 \cup (A_2 \cap A_3) \]This captures either event: site 1 completed or sites 2 and 3 completed together.

07

Drawing and Shading the Venn Diagram

Create a Venn diagram with three overlapping circles, each representing one of \( A_1 \), \( A_2 \), and \( A_3 \). Shade the relevant areas for each scenario:- Part (a): Shade all areas covered by any circle.- Part (b): Shade only the region where all three circles overlap.- Part (c): Shade only the region for \( A_1 \) that does not intersect with \( A_2 \) or \( A_3 \).- Part (d): Shade non-overlapping parts of each circle separately.- Part (e): Shade \( A_1 \) and the intersection of \( A_2 \) and \( A_3 \).

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Key Concepts

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

Union of Events

The union of events in probability refers to the occurrence of at least one of several events. Think of it as the combination of all possible outcomes where any event is true. In mathematical terms, if we have events \(A_1, A_2, A_3\), their union is represented by \(A_1 \cup A_2 \cup A_3\).

• This denotes the situation where at least one plant is completed by the contract date.
• The union operation is inclusive, encompassing all cases where any single event or multiple events occur.

When you hear 'at least one,' that's a trigger to think about union of events. In Venn diagrams, this would involve shading all regions covered by any of the circles representing the events.

Intersection of Events

The intersection of events is where multiple events happen together, meaning all must occur at the same time. Using our events, this is symbolized by \(A_1 \cap A_2 \cap A_3\).

• This signifies that all plants are completed by the contract date.
• The intersection represents the overlap, the part where all conditions of the involved events are satisfied.

Imagine a Venn diagram: the intersection is the shared area where all circles overlap. Intersections demand a stricter condition because they require every event involved to happen simultaneously. It's ideal for scenarios where complete success or consensus is needed.

Complementation of Events

Complementation involves outcomes where an event does not occur. If we consider the event \(A_i\) where a plant at site \(i\) is completed, \(A_i^c\) is its complement, meaning it is not completed.

• A complementation is a defining feature of expressing only one event's occurrence among others, like completing only the plant at site 1, which would be \(A_1 \cap A_2^c \cap A_3^c\).

In Venn diagrams, this would mean shading parts that reflect the occurrence of the intended event while explicitly excluding others from happening. Complements are useful to express exclusivity or lack of occurrence.

Venn Diagram

Venn diagrams are visual representations that make it easy to understand relationships between different events in probability. They consist of circles that represent events and their overlaps show intersections.

• To depict at least one plant being completed, shade all circles.
• For all plants completed, shade only the central overlapping area.
• Only one plant's completion, such as site 1, translates to shading its circle while excluding others.

This tool beautifully depicts complex relations like unions, intersections, and complements, making abstract concepts more tangible. It is especially helpful for visual learners as it shows exactly how events overlap or exclude one another.

Conditional Probability

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. It's like asking, "What's the probability of event A happening if event B is known to occur?".

• Conditional probability formula: \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\) assuming \(P(B) > 0\).
• This concept often combines different events and is useful when events influence or depend on each other.

While not directly featured in the given scenarios, it's an important aspect of understanding how occurrences interplay within dependent constructs. Solid grasp on conditional probability helps contextualize real-life situations dependent on other events.