Problem 35 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Robin purchased shares of a machine tool company and shares of an airline company. Let \(E\) be the event that the shares of the machine tool company increase in value over the next \(6 \mathrm{mo}\), and let \(F\) be the event that the shares of the airline company increase in value over the next \(6 \mathrm{mo}\). Using the symbols \(\cup, \cap\), and \({ }^{c}\), describe the following events. a. The shares in the machine tool company do not increase in value. b. The shares in both the machine tool company and the airline company do not increase in value. c. The shares of at least one of the two companies increase in value. d. The shares of only one of the two companies increase in value.

Step-by-Step Solution

Answer

a. \(E^c\) b. \(E^c \cap F^c\) c. \(E \cup F\) d. \(E \cap F^c \cup F \cap E^c\)

Step 1: Event A: Shares in the machine tool company do not increase in value#

We are given event \(E\) as the increase in the value of the machine tool company's shares. To describe the event that the shares in the machine tool company do not increase in value, we simply need to represent the complement of event \(E\). That is: \(E^c\)

Step 2: Event B: Shares in both companies do not increase in value#

We need to describe the event in which both the shares of the machine tool company (event \(E\)) and the airline company (event \(F\)) do not increase in value. Since we are considering both events occurring simultaneously, we will use the intersection symbol (\(\cap\)). But we need the complement of both events, so we have: \(E^c \cap F^c\)

Step 3: Event C: Shares of at least one of the two companies increase in value#

This event is asking for the occurrence of either event \(E\) (increase in the value of the machine tool company's shares) or event \(F\) (increase in the value of the airline company's shares), or both. We use the union symbol (\(\cup\)) to represent this event: \(E \cup F\)

Step 4: Event D: Shares of only one of the two companies increase in value#

In this event, we are looking for a situation where only one of the events \(E\) or \(F\) occurs, but not both. We can describe this event by creating two separate scenarios: event \(E\) occurs while event \(F\) does not (using the complement \(F^c\)), and event \(F\) occurs while event \(E\) does not (using the complement \(E^c\)). Since we are considering the combination of these two scenarios as a single event, we need to use the union symbol (\(\cup\)). The event can be described as: \(E \cap F^c \cup F \cap E^c\)

Key Concepts

Event Complements

In probability, the complement of an event is essentially what the event does not account for. If an event is denoted as \(E\), its complement is represented as \(E^c\). It refers to the scenario where event \(E\) does not occur. For instance, in our example where \(E\) is the event of an increase in share value of a machine tool company over six months, \(E^c\) indicates that the share value does not increase in that time frame.
The concept of event complements is crucial in probability because it helps us understand all possible outcomes in a simpler way. Every event has its complement, and together, these covers all possible scenarios, encompassing 100% probability. Recognizing this can make complex probability questions easier to tackle.

Union Symbol in Probability

The union symbol, represented as \(\cup\), is used in probability to denote the occurrence of at least one of multiple events. It effectively combines two or more events into one larger event that accounts for the occurrence of any of the listed events.
In our exercise, \(E \cup F\) represents the event where either the machine tool company's shares or the airline company's shares (or both) increase in value within six months. This is like saying we're considering the success of at least one company as a favorable outcome.
Understanding the union symbol is essential when dealing with probabilities involving multiple potential outcomes. It allows us to broaden the scope and grasp a situation comprehensively with all alternative successful outcomes considered.

Intersection Symbol in Probability

The intersection symbol in probability is represented as \(\cap\). It signifies the occurrence of two or more events occurring at the same time. When two events intersect, it means all the listed events must happen simultaneously.
In our case, to express that neither the machine tool company's shares nor the airline company's shares increase in value, we use \(E^c \cap F^c\). This intersection means that both complements occur, indicating both events do not happen.
Grasping the intersection of events helps in understanding complex probabilities where multiple conditions must be met together. It narrows down all scenarios to just those that genuinely meet all specified criteria, crucial for precise probability assessments.