Problem 6 - 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. Are the events \(E \cup F\) and \(E \cap F^{c}\) mutually exclusive?

Step-by-Step Solution

Answer

The events \(E \cup F\) and \(E \cap F^{c}\) are not mutually exclusive since their intersection \((E \cup F) \cap (E \cap F^{c}) = \{a, b, d, f\} \cap \{b\} = \{b\}\) is not an empty set.

Step 1: Calculate \(E \cup F\) event

To calculate \(E \cup F\), we need to find the union of events E and F, which includes all elements that are in either E or F or both of them. \(E \cup F = \{a, b\} \cup \{a, d, f\} = \{a, b, d, f\}\)

Step 2: Calculate \(F^{c}\) (the complement of F)

To find the complement of F, denoted by \(F^{c}\), we need to find all elements in the sample space S that are not in F. \(F^{c} = S - F = \{a, b, c, d, e, f\} - \{a, d, f\} = \{b, c, e\}\)

Step 3: Calculate \(E \cap F^{c}\) event

To calculate the intersection of events E and \(F^{c}\), denoted by \(E \cap F^{c}\), we need to find the elements that are common to both E and \(F^{c}\). \(E \cap F^{c} = \{a, b\} \cap \{b, c, e\} = \{b\}\)

Step 4: Check if \(E \cup F\) and \(E \cap F^{c}\) are mutually exclusive

Now, we need to check if the intersection of events \(E \cup F\) and \(E \cap F^{c}\) is an empty set. \((E \cup F) \cap (E \cap F^{c}) = \{a, b, d, f\} \cap \{b\} = \{b\}\) Since the intersection is not an empty set, the events \(E \cup F\) and \(E \cap F^{c}\) are not mutually exclusive.

Key Concepts

Sample Space

In probability theory, a sample space is the set of all possible outcomes of an experiment. When you are dealing with an experiment, like flipping a coin or rolling a die, the sample space includes every possible result that could occur from that action.

For instance, if you roll a six-sided die, the sample space is \( \{1, 2, 3, 4, 5, 6\} \). Each of the numbers represents an outcome of the die. In our given exercise, the sample space \( S \) is \( \{a, b, c, d, e, f\} \), which signifies all the elements listed are all possible outcomes of this specific experiment.

Each subset of the sample space is called an "event." Events are what we evaluate to understand probability, and they are central to predicting outcomes in probability theory.

Mutually Exclusive Events

Events are considered mutually exclusive if they cannot occur at the same time. This means there is no overlap between the events. If you roll a single six-sided die, the events "rolling an odd number" and "rolling an even number" are mutually exclusive because a number cannot be both odd and even simultaneously.

In our exercise, we calculated \((E \cup F)\) which is \( \{a, b, d, f\} \), and \((E \cap F^{c})\) which is \( \{b\} \).

If these were mutually exclusive events, they would have no common elements, meaning their intersection \((E \cup F) \cap (E \cap F^{c})\) would be an empty set. However, since the intersection contains \(\{b\}\), we can ascertain that these events are not mutually exclusive.

Event Complement

The concept of an event complement involves considering all outcomes that are not part of the event of interest. Given a sample space \( S \) and an event \( F \), the complement of \( F \), denoted \( F^{c} \), is the set of outcomes in the sample space that are not in \( F \).

Imagine a full deck of cards. If your event \( F \) was drawing a heart, then \( F^{c} \) includes all the spades, diamonds, and clubs. In our example, the sample space \( S \) is \( \{a, b, c, d, e, f\} \) and \( F \) is \( \{a, d, f\} \). Thus, the complement \( F^{c} \) consists of \( \{b, c, e\} \) since these outcomes aren't part of \( F \).

Understanding the complement is crucial for determining probabilities linked with not happening of an event.

Set Operations

Set operations are mathematical actions applied to sets, and they are essential in probability theory as they help in evaluating combinations of events. The primary set operations in probability include union, intersection, and complementation.

• Union (\(\cup\)): This operation combines all elements that are in either of the sets or in both. In our exercise, \(E \cup F\) gave us the combination of all elements from set \(E\) and set \(F\).


• Intersection (\(\cap\)): This includes only the elements that are common between sets. For instance, \(E \cap F^{c}\) gave us set \(\{b\}\) which is the common element in both \(E\) and \(F^{c}\).


• Complementation (\((\cdot)^{c}\)): This includes all the elements that are in the universal set (here, the sample space) but not in the set in question. \(F^{c}\) was calculated as all elements in \(S\) that are not in \(F\).

Using these operations effectively helps in analyzing different event scenarios and aids in solving complex probability problems.