Question

A sample space consists of \(S=\left\\{E_{1}, E_{2}\right.\), \(\left.E_{3}, E_{4}\right\\} .\) List the simple events in "both \(A\) and \(B\)," " \(A\) or \(B\) or both," and "not \(B\) " for the events given in Exercises \(13-15 .\) $$A=\left\\{E_{2}, E_{4}\right\\} \text { and } B=\left\\{E_{2}, E_{3}, E_{4}\right.$$

Short answer

1. A and B 2. A or B or both 3. not B Answer: 1. A and B: {E2, E4} 2. A or B or both: {E2, E3, E4} 3. not B: {E1}

Step-by-step solution

Find the Intersection of A and B (A ∩ B)

The intersection of two events A and B is the set of all simple events that are in both A and B. Given the events: $$ A = \{E_2, E_4\} \text{ and } B = \{E_2, E_3, E_4\} $$ We compare the simple events in A and B and find the common elements. In this case, the common elements (simple events in both A and B) are \(E_2\) and \(E_4\). So, the intersection of A and B (A ∩ B) is: $$ A \cap B = \{E_2, E_4\} $$

Find the Union of A and B (A ∪ B)

The union of two events A and B is the set of all simple events that are in either A or B or both. Given the events: $$ A = \{E_2, E_4\} \text{ and } B = \{E_2, E_3, E_4\} $$ We combine the simple events in A and B, removing any duplicate elements. In this case, the union of A and B (A ∪ B) is: $$ A \cup B = \{E_2, E_3, E_4\} $$

Find the Complement of B (B')

The complement of an event B (denoted as B') is the set of all simple events in the sample space S that are not in B. Given the sample space S and event B: $$ S = \{E_1, E_2, E_3, E_4\} \text{ and } B = \{E_2, E_3, E_4\} $$ We identify the simple events in S that are not in B. In this case, the simple event \(E_1\) is not in B. So, the complement of B (B') is: $$ B' = \{E_1\} $$ In conclusion, we have found the simple events in the following cases: 1. A and B: \(\{E_2, E_4\}\) 2. A or B or both: \(\{E_2, E_3, E_4\}\) 3. not B: \(\{E_1\}\)

Key concepts

Simple Events

In probability theory, a simple event is an outcome that cannot be broken down into smaller parts. It represents a single outcome in the context of all possible outcomes, which together comprise the sample space. For example, when rolling a six-sided die, each possible roll (1, 2, 3, 4, 5, 6) is considered a simple event.

In the provided exercise, the sample space of possible events is defined as S = \(\big\{E_{1}, E_{2}, E_{3}, E_{4}\big\}\). Within this space, any individual outcome, like \(E_{2}\) or \(E_{4}\), is a simple event since it cannot be decomposed further. Understanding simple events is fundamental as they are the building blocks of more complex events in the realm of probability.

Intersection of Sets

When working with probabilities, the intersection of two sets is a crucial concept. Denoted by the symbol \(\cap\), the intersection represents all elements that two or more sets have in common. In the context of probability, these are the outcomes that are possible in all the sets being considered.

In the exercise, we have the sets A = \(\big\{E_{2}, E_{4}\big\}\) and B = \(\big\{E_{2}, E_{3}, E_{4}\big\}\). To find their intersection, you look for simple events that appear in both A and B. This would be \(A \cap B = \big\{E_{2}, E_{4}\big\}\) as both simple events are shared by A and B. This concept is essential for understanding how to determine the likelihood of two conditions happening simultaneously.

Union of Sets

The union of sets, indicated by the symbol \(\cup\), combines all the elements from the involved sets into one set, removing any duplicates. It's the set of outcomes that occur in either of the sets or in both. In probability, the union of events represents the total set of possible outcomes where at least one of the conditions is satisfied.

For our example, the union of sets A and B, which are A = \(\big\{E_{2}, E_{4}\big\}\) and B = \(\big\{E_{2}, E_{3}, E_{4}\big\}\), includes all simple events that are part of either A or B, thus it's \(A \cup B = \big\{E_{2}, E_{3}, E_{4}\big\}\). This operation is foundational for calculating the probability of at least one of several events occurring.

Complement of a Set

In probability, the complement of a set A, often denoted as A', consists of all the elements in the sample space S that are not in A. It's the opposite of the set, comprising all the outcomes that do not satisfy the condition described by set A.

In the exercise, the complement of set B (B = \(\big\{E_{2}, E_{3}, E_{4}\big\}\)) with respect to the sample space S = \(\big\{E_{1}, E_{2}, E_{3}, E_{4}\big\}\) is found by identifying what is in S but not in B. The result is B' = \(\big\{E_{1}\big\}\), which includes only the simple event \(E_{1}\), as it is the sole element not present in B. Recognizing how to find the complement is important for determining the probability of an event not happening.