Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(S\) be any sample space and let E, \(\boldsymbol{F}\), and \(\boldsymbol{G}\) be any three events associated with the experiment. Describe the events using the symbols \(\cup, \cap\), and . The event that \(G\) does not occur

Step-by-Step Solution

Answer

The event where G does not occur can be represented using the complement symbol as \(G^c\) or \(G'\).

Step 1: Understand the symbols and notation

In set theory, the symbols used are: - \(\cup\) : Union of two sets (all elements that are in either set) - \(\cap\) : Intersection of two sets (all elements that are in both sets) - \(^c\) or \('\) : Complement of a set (all elements that are NOT in the specified set) In this case, the events E, F, and G are all subsets of the sample space S.

Step 2: Represent the event where G does not occur

To represent the event where G does not occur, we use the complement symbol. So, the event where G does not occur is represented by: \(G^c\) or \(G'\) Both notations can be used to describe the complement of G, meaning that G does not occur in the given sample space S.

Key Concepts

Sample Space

In probability theory, the concept of a sample space is fundamental. A sample space, denoted as \( S \), includes all possible outcomes of an experiment or a random process. For instance, when tossing a coin, the sample space is \( \{\text{heads}, \text{tails}\} \). Every event that occurs during this experiment must be a subset of the sample space.
Understanding the sample space is crucial because it establishes the context for defining events and calculating probabilities. It acts as a universal set while considering the probability of different outcomes or events in an experiment.

• Example 1: The sample space for a six-sided die roll is \( \{1, 2, 3, 4, 5, 6\} \). Each roll's outcome is a subset of this space.
• Example 2: In a deck of cards, the sample space for drawing one card is the set of all 52 possible cards.

Clearly defining your sample space at the beginning of solving probability problems helps ensure accuracy and clarity throughout the process.

Set Theory

The backbone of probability is built on set theory, which deals with the collection of objects or elements. These elements could be anything from numbers to symbols to entire events. Set theory is the language used to describe events and their relationships in probability.

• Intersection (\( \cap \)): Represents events occurring together. For example, the intersection of Event E and Event F, denoted as \( E \cap F \), includes all outcomes common to both.
• Union (\( \cup \)): Refers to at least one of multiple events occurring. So, \( E \cup F \) includes any outcome in either Event E or F.
• Complement: Describes the occurrence when a specific event does not occur. It profoundly relates to understanding events' probability and eventual outcomes.

Set theory allows us to formalize and solve complex probability problems by representing events systematically and manipulating them using set operations.

Complement of a Set

The complement of a set captures the idea of 'not happening' within probability theory. Symbolically, if \( G \) is an event, then \( G^c \) or \( G' \) represents the complement of \( G \). This complement includes all elements of the sample space \( S \) that are not in \( G \).
The complement is essential for determining the probability of an event not occurring. If you know the probability of \( G \) (say \( P(G) \)), the probability of not \( G \) (i.e., \( G^c \)) is calculated as:\[P(G^c) = 1 - P(G)\]Understanding the complement helps in various real-world applications such as error estimation, where determining what doesn’t happen is just as informative as what does.

• Example: If the probability of raining on a given day is 0.3, the probability that it will not rain is 0.7, which is the complement.

In sum, mastering the complement concept allows for comprehensive probability analyses and improved decision-making in uncertain scenarios.