Question

A sample space contains seven simple events: \(E_{1}, E_{2}, \ldots, E_{7} .\) Use the following three eventsA, \(B\), and \(C\) - and list the simple events in Exercises \(7-12\). \(A=\left\\{E_{3}, E_{4}, E_{6}\right\\} \quad B=\left\\{E_{1}, E_{3}, E_{5}, E_{7}\right\\} \quad C=\left\\{E_{2}, E_{4}\right\\}\) $$A \text { or } C \text { or both }$$

Short answer

Answer: The simple events that belong to event A or C or both are: \(E_2, E_3, E_4, E_6\).

Step-by-step solution

Understand the Given Events

We are given three events A, B, and C with their corresponding simple events. Event A contains simple events \(E_3, E_4, E_6\). Event B contains simple events \(E_1, E_3, E_5, E_7\). Event C contains simple events \(E_2, E_4\).

Finding the Union of A and C

To find the simple events that belong to A or C or both, we need to find the union of A and C, denoted by \(A \cup C\). The union of two events is the set of all simple events that belong to either A, C, or both.

Determine the Union of A and C

List the simple events of \(A \cup C\) without duplication. \(A \cup C = \left\{ E_2, E_3, E_4, E_6\right\}\)

Final Answer

The simple events in \(A \text { or } C \text { or both }\) are: \(A \cup C = \left\{ E_2, E_3, E_4, E_6\right\}\).

Key concepts

Sample Space

A sample space in probability refers to the set of all possible outcomes or events that can happen in a given experiment. In the context of our exercise, the sample space is comprised of seven simple events: \( E_1, E_2, \ldots, E_7 \). Think of it as a comprehensive list that includes every potential result of the scenario you are considering.

When you're dealing with events, always start by identifying and understanding the sample space. This is because every event, whether simple or compound, will be a subset of the sample space.

In simpler terms, without understanding what the complete set of possible outcomes is, you cannot effectively determine the probability or occurrence of any given event.

Union of Events

The union of events is a fundamental concept in probability that reflects the outcomes occurring in at least one of multiple events. Mathematically, if you have two events, say \( A \) and \( C \), the union is expressed as \( A \cup C \). This encompasses all simple events that belong to either \( A \), \( C \), or both.

In our exercise, we had events \( A \) and \( C \) given as:

• \( A = \{E_3, E_4, E_6\} \)
• \( C = \{E_2, E_4\} \)

To find the union, simply combine all distinct events from both sets: \( A \cup C = \{ E_2, E_3, E_4, E_6 \} \). Notice that if an event, like \( E_4 \), occurs in both \( A \) and \( C \), it only appears once in the union.

Understanding the union of events helps in solving problems related to determining the likelihood of multiple scenarios occurring.

Simple Events

Simple events are the most basic outcomes that cannot be decomposed further. Each simple event represents a single point in the sample space.

In the context of our problem, each \( E_i \) (e.g., \( E_1, E_2, \ldots, E_7 \)) serves as a simple event. These are the building blocks for more complex events.

When analyzing simple events, remember:

• Each one stands alone and represents one specific outcome.
• All simple events together make up the whole sample space.
• Simple events can combine to form more complex event sets, such as the union of events.

By understanding each simple event, you can better interpret the dynamics and probabilities of more intricate combinations of events like \( A \), \( B \), and \( C \).

Event Analysis

Event analysis in probability involves breaking down complex events into simpler, understandable components to determine the likelihood of different scenarios.

In our exercise scenario, event analysis begins by identifying the relevant simple and compound events that make up the given sets \( A \), \( B \), and \( C \). It is crucial to:

• Start with understanding each simple event and its place within the sample space.
• Use operations such as union, intersection, and complement to analyze compound events.
• Determine which events occur within a specific scenario, as done for \( A \cup C \).

With this analytical mindset, you efficiently solve probability problems by comprehending the relationships and operations between different events. This approach helps demystify complex scenarios, making it easier to calculate probabilities accurately.