Problem 3 - 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 \(F^{c}\) and \(E \cap G^{c}\).

Step-by-Step Solution

Answer

The events \(F^{c}\) and \(E \cap G^{c}\) are \(\{b, c, e\}\) and \(\{a\}\), respectively.

Step 1: Find the complement of event F (\(F^{c}\))

To find the complement of event F, we need to take the elements in the sample space S that are not in F. In this case, \(F=\{a, d, f\}\). The elements in the sample space S but not in F are the complement of F. \(F^{c} = S \setminus F = S \cap F^{c} = \{a, b, c, d, e, f\} \cap \{a, d, f\}^{c}\) Comparing the elements, we find that: \(F^{c} = \{b, c, e\}\)

Step 2: Find the complement of event G (\(G^{c}\))

To find the complement of event G, we take the elements in the sample space S that are not in G. In this case, \(G=\{b, c, e\}\). The elements in the sample space S but not in G are the complement of G. \(G^{c} = S \setminus G = S \cap G^{c} = \{a, b, c, d, e, f\} \cap \{b, c, e\}^{c}\) Comparing the elements, we find that: \(G^{c} = \{a, d, f\}\)

Step 3: Find the intersection of event E and the complement of event G (\(E \cap G^{c}\))

Now we have the events E and \(G^{c}\), and we need to find their intersection. The intersection of two events contains the elements common to both events. In this case, \(E = \{a, b\}\) and \(G^{c} = \{a, d, f\}\). \(E \cap G^{c} = \{a, b\} \cap \{a, d, f\}\) Comparing the elements, we find that: \(E \cap G^{c} = \{a\}\) So, the events \(F^{c}\) and \(E \cap G^{c}\) are \(\{b, c, e\}\) and \(\{a\}\), respectively.

Key Concepts

Sample Space

In probability theory, the **sample space** of an experiment is a fundamental concept that denotes the set of all possible outcomes. Each individual outcome is known as a sample point. In our exercise, the sample space is denoted by the set \(S = \{a, b, c, d, e, f\}\), which means our experiment could result in any of these six outcomes. For example, this could represent a situation where six different events or states are possible, and these are labeled from 'a' to 'f'.
Understanding the sample space is crucial because any event related to the experiment must be a subset of this space. This allows us to systematically evaluate probabilities and determine event relationships. The sample space forms the basis upon which we identify other important concepts like complements and intersections of events.

Complement of an Event

The **complement of an event** in probability theory is the set of outcomes that are not part of the event in question. If an event \(X\) consists of a subset of the sample space \(S\), then the complement of \(X\), denoted by \(X^c\), consists of all elements in \(S\) that are not in \(X\).
In the exercise, we first find the complement of the event \(F\), which was \(F = \{a, d, f\}\). The elements not in \(F\) from the sample space \(S\) are \(\{b, c, e\}\). Thus, \(F^c = \{b, c, e\}\).
Next, we find the complement of \(G\) which is \(\{b, c, e\}\), and the elements not in \(G\) from the sample space \(S\) are \(\{a, d, f\}\). Hence, \(G^c = \{a, d, f\}\).
Understanding complements is vital for calculating probabilities, as they help identify what 'not happening' looks like for any event.

Intersection of Events

In probability, the **intersection of events** refers to a new event that contains all the outcomes that two or more events share. The intersection is denoted by the symbol \( \cap \).
When we have two events \(E\) and \(F\), their intersection \(E \cap F\) represents all the outcomes that appear in both event \(E\) and event \(F\). This concept helps find common probabilities.
In our exercise, we calculated the intersection of event \(E\) with the complement of event \(G\). Given that \(E = \{a, b\}\) and \(G^c = \{a, d, f\}\), the intersection is made up of elements found in both sets, resulting in \(E \cap G^c = \{a\}\).
The intersection of events is key in understanding shared probabilities or the simultaneous occurrence of multiple events, making it a powerful tool in probability calculations.