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 \(7-12\). \(A=\left\\{E_{3}, E_{4}, E_{6}\right\\} \quad B=\left\\{E_{1}, E_{3}, E_{5}, E_{7}\right\\} \quad C=\left\\{E_{2}, E_{4}\right\\}\) $$\text { Both } A \text { and } B$$

Short answer

Answer: The intersection of events A and B is given by {E₃}.

Step-by-step solution

Identify the elements in event A

First, let's list the elements in event \(A\). According to the problem, \(A=\left\\{E_{3}, E_{4}, E_{6}\right\\}\).

Identify the elements in event B

Next, let's list the elements in event \(B\). According to the problem, \(B=\left\\{E_{1}, E_{3}, E_{5},E_{7}\right\\}\).

Find the intersection of events A and B

Now, we want to find the simple events that are present in both events \(A\) and \(B\). This is called the intersection of events \(A\) and \(B\), denoted as \(A \cap B\). To find this intersection, we compare the elements in event \(A\) to the elements in event \(B\) and list the elements that are common to both events.

List the elements in the intersection of events A and B

Upon comparison, we can see that \(E_{3}\) is the only element common to both events \(A\) and \(B\). Therefore, the intersection of events \(A\) and \(B\) is given by: $$A \cap B = \left\\{E_{3}\right\\}$$

Key concepts

Sample Space

A sample space is a fundamental concept in probability theory that represents the set of all possible outcomes of a random experiment. Think of it as a complete list of every potential result that can be attained.
For example, when you roll a six-sided die, the sample space is \[S = \{1, 2, 3, 4, 5, 6\}\]. Each result from rolling the die represents one possible outcome within this sample space.
In our exercise, the sample space contains seven elements: \(E_{1}, E_{2}, \ldots, E_{7}\), representing all the simple events that might occur. Recognizing the sample space is crucial because it allows us to consider all potential outcomes and make informed probabilistic predictions.

Intersection of Events

The intersection of events in probability refers to the scenario where two or more events occur at the same time. It's a way to determine which outcomes are common to all the events being considered.
Mathematically, the intersection is denoted by the symbol \(\cap\). For instance, if we have two events \(A\) and \(B\), the intersection \(A \cap B\) includes only the outcomes that are present in both \(A\) and \(B\).
In our exercise, we're asked to find the intersection of events \(A\) and \(B\) from our sample space. By identifying which simple events occur in both \(A\) and \(B\), we determine that \(A \cap B = \{E_3\}\). This indicates that \(E_3\) is the only outcome they share in common.

Simple Events

Simple events, in the context of probability, are the most basic possible results of a random experiment. Each simple event corresponds to a single outcome and cannot be further broken down into more simple components.
For example, when tossing a coin, the simple events are "heads" and "tails". These are the basic outcomes of the action.
In our sample space \(\{E_1, E_2, \ldots, E_7\}\), each \(E_i\) represents a simple event. These form the building blocks that combine to establish more complex events, like event \(A = \{E_3, E_4, E_6\}\), which is simply a collection of three simple events.

Events in Probability

Events in probability are sets of outcomes from a sample space, which can be one or several simple events. An event can be as basic as a single outcome or as complex as any combination of multiple outcomes.
Using our exercise, event \(A = \{E_3, E_4, E_6\}\) and event \(B = \{E_1, E_3, E_5, E_7\}\) are examples of events composed of simple events from the sample space.
Events are significant because they are the basic components of probability calculations. The probability of an event is found by considering the ratio of the number of outcomes in the event to the total number of outcomes in the sample space. These events can also interact in various ways, such as through intersections or unions, to help analyze different probabilistic scenarios.