Problem 18 - 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 \(E\) but not \(F\) occurs

Step-by-Step Solution

Answer

The event that \(E\) occurs but not \(F\) can be described using the given symbols as: \[E \cap F^c\]

Step 1: Identify the desired event

The desired event is that \(E\) occurs, but \(F\) does not occur. This means we want to consider the region where \(E\) and the complement of \(F\) (which represents the event that \(F\) does not occur) are overlapping.

Step 2: Use intersection symbol to represent overlapping events

To represent the overlapping of two events (in this case, \(E\) and the complement of \(F\)), we use the intersection symbol (\(\cap\)). So, we write \(E \cap F^c\) to denote the event that \(E\) occurs but not \(F\).

Step 3: Final answer

The event that \(E\) occurs but not \(F\) can be described using the given symbols as: \[E \cap F^c\]

Key Concepts

Sample Space

Imagine you are playing a board game where you roll a single six-sided die. The sample space for this experiment is the set of all possible outcomes that can occur when the die is rolled. In probability theory, this concept is fundamental. It is denoted by the symbol \( S \) and represents the universal set of all possible outcomes of a random experiment.

For instance, the sample space for our die roll would be \( S = \{1, 2, 3, 4, 5, 6\} \). No matter what roll you're hoping for, it must be within this set. It's critical to define the sample space correctly because it sets the stage for determining the probability of various events.

Events in Probability

Within the sample space, we can define events, which are simply subsets of the sample space that may occur as result of the experiment. In our die example, an event might be rolling an even number. This event can be represented as \( E = \{2, 4, 6\} \).

An important aspect of events in probability is that they can be composed of multiple outcomes and that we can assign probabilities to these events. The probability of an event is the sum of the probabilities of the outcomes within that event. Understanding how events are defined and how they relate to the sample space is essential for solving probability problems.

Union and Intersection of Sets

In probability, operations such as union and intersection are used to describe combinations of events. The union of two events, denoted by \( \cup \), represents the event that at least one of the events occurs. Conversely, the intersection of events, denoted by \( \cap \), represents the event that both events occur simultaneously.

For example, if we continued with our dice game and defined a new event \( F \) as rolling a number greater than 4, which is \( F = \{5, 6\} \), then the union of events \( E \) and \( F \) would be the set \( E \cup F = \{2, 4, 5, 6\} \), representing all even numbers or any number greater than 4. The intersection would be \( E \cap F = \{6\} \), since 6 is both an even number and greater than 4. These concepts are exceedingly useful for understanding complex probabilities and are directly applicable to the textbook exercise in identifying the likelihood of combined events.