Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(S=\\{a, b, c, d, e, f\\}\) be a sample space of an experiment and let \(E=\\{a, b\\}, F=\\{a, d, f\\}\), and \(G=\\{b, c, e\\}\) be events of this experiment. Find the events \(F \cup G\) and \(F \cap G\).

Step-by-Step Solution

Answer

The union of events \(F\) and \(G\) is \(F \cup G = \{a, b, c, d, e, f\}\), and the intersection of events \(F\) and \(G\) is \(F \cap G = \emptyset\).

Step 1: Write down the given information

We are given that the sample space \(S = \{a, b, c, d, e, f\}\), and the events \(E = \{a, b\}\), \(F = \{a, d, f\}\), and \(G = \{b, c, e\}\).

Step 2: Find the union of events F and G, denoted F ∪ G

To find the union of events \(F\) and \(G\), we combine all the unique outcomes of each event, i.e., all outcomes that belong to either \(F\) or \(G\). This can be written as: \[F \cup G = \{a, d, f\} \cup \{b, c, e\}\] Next, we list all the unique outcomes of \(F\) and \(G\): \[F \cup G = \{a, b, c, d, e, f\}\]

Step 3: Find the intersection of events F and G, denoted F ∩ G

To find the intersection of events \(F\) and \(G\), we look for the outcomes that belong to both \(F\) and \(G\). This can be written as: \[F \cap G = \{a, d, f\} \cap \{b, c, e\}\] Since there are no common elements in the two sets, the intersection of \(F\) and \(G\) is an empty set: \[F \cap G = \emptyset\]

Step 4: Write down the final answer

The union of events \(F\) and \(G\) is: \[F \cup G = \{a, b, c, d, e, f\}\] The intersection of events \(F\) and \(G\) is: \[F \cap G = \emptyset\]

Key Concepts

Sample Space

In the world of probability, a **sample space** is the set of all possible outcomes of an experiment. It is a fundamental concept as it lays the groundwork for studying events and their probabilities. In our exercise, the sample space is represented as:

• \(S = \{a, b, c, d, e, f\}\)

This means that when you perform the experiment, the outcome will be one of these elements. Each element in this set is considered an outcome. When experimenting, every possible result must appear in the sample space.

For clarity, think about rolling a die. The sample space would be \(\{1, 2, 3, 4, 5, 6\}\), representing all possible numbers that could land on top. Similarly, our sample space \(S\) relates to an experiment that could result in the outcomes \(a, b, c, d, e,\) or \(f\). A well-defined sample space ensures that we correctly analyze and understand different possible events within an experiment.

Union of Events

The **union of events** refers to the combination of two or more events in a way that includes all outcomes that belong to either event. In probabilistic terms, if you want to know the probability of either event 'A' or event 'B' occurring, you use the union of those events, denoted as \(A \cup B\). In our problem, the events \(F\) and \(G\) are given as:

• \(F = \{a, d, f\}\)
• \(G = \{b, c, e\}\)

To find the union \(F \cup G\), we simply merge all unique elements from both sets:

• \(F \cup G = \{a, d, f\} \cup \{b, c, e\}\)
• \(F \cup G = \{a, b, c, d, e, f\}\)

The result is a set that includes every outcome from both \(F\) and \(G\). This makes sense because the combination of these events allows for any result that occurs in \(F\), \(G\), or both. This approach helps us understand the occurrence of any one of the events without considering overlaps since, in unions, we treat duplicates as one entry in the resulting set.

Intersection of Events

An **intersection of events** captures outcomes that two sets have in common. When looking to calculate the intersection of events, you're interested in occurrences that are both in event 'A' and event 'B'. The intersection is symbolically denoted as \(A \cap B\). In our specific example, we have:

• \(F = \{a, d, f\}\)
• \(G = \{b, c, e\}\)

To discover the intersection \(F \cap G\), we identify common elements:\[F \cap G = \{a, d, f\} \cap \{b, c, e\}\]After comparing the elements in both sets, we find:\[F \cap G = \emptyset\]This indicates that there are no elements common to both sets \(F\) and \(G\), resulting in an empty set for their intersection. Understanding intersections helps us determine the probability of two events occurring simultaneously, which can be critical, particularly in complex probability scenarios where joint occurrence matters.