Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(S=\\{a, b, c\\}\) be a sample space of an experiment with outcomes \(a, b\), and \(c\). List all the events of this experiment.

Step-by-Step Solution

Answer

The possible events of the experiment are: \(\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\}\).

Step 1: Understand the sample space and event concept

The sample space, in this case, consists of three possible outcomes: \(a, b, c\). An event is any subset of the sample space, which can include no outcomes, one outcome, two outcomes, or all three outcomes.

Step 2: List events with zero outcomes

The first event subset will contain no outcomes, representing an event that none of the outcomes (\(a, b, c\)) occurs. This is represented by the empty set, denoted by \(\emptyset\).

Step 3: List events with exactly one outcome

Next, we will list the events that have exactly one outcome. There are three such events: 1. The event containing only outcome \(a\): \(\{a\}\) 2. The event containing only outcome \(b\): \(\{b\}\) 3. The event containing only outcome \(c\): \(\{c\}\)

Step 4: List events with exactly two outcomes

Now, we will list the events that have exactly two outcomes. There are three such events: 1. The event containing outcomes \(a\) and \(b\): \(\{a, b\}\) 2. The event containing outcomes \(a\) and \(c\): \(\{a, c\}\) 3. The event containing outcomes \(b\) and \(c\): \(\{b, c\}\)

Step 5: List events with all three outcomes

Finally, we have the event with all three outcomes: \(\{a, b, c\}\).

Step 6: Combine all events to form the power set

Now that we have listed the events with zero, one, two, and three outcomes, we can combine them to form the power set of the given sample space. The power set of set \(S = \{a, b, c\}\) is: \[P(S) = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\}\}\] These are all the possible events of the given experiment.

Key Concepts

Sample Space

The concept of sample space is fundamental in the study of probability. It is defined as the set of all possible outcomes of an experiment. For instance, if we toss a coin, the sample space would be \(S = \{Heads, Tails\}\), as these are all the potential results that could occur. In the given exercise, the sample space for the experiment is \(S = \{a, b, c\}\), indicating that there are three distinct outcomes that could happen as a result of the experiment.
Understanding the sample space is critical because it sets the stage for determining the probability of various events. An event refers to any combination of outcomes from the sample space, and the probability of an event is a measure of how likely it is to occur.
Thus, by clearly understanding the sample space, students can better identify and evaluate the possible events and their probabilities, which is the cornerstone of any probabilistic analysis.

Power Set

The power set is a term used to describe the set of all subsets of a given set, including the set itself and the empty set. This concept is especially important in probability, as the power set represents all possible events that could occur in an experiment. Imagine a power set as a catalog of all potential outcomes, no matter what the experiment is.
In our exercise, the sample space is \(S = \{a, b, c\}\), and the power set denoted by \(P(S)\) would include the empty set (representing an event where nothing occurs), all individual outcomes, all pairs of outcomes, and the set of all outcomes itself. Hence, the power set for our sample space is \[P(S) = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\}\}\].
The calculation of the power set is crucial for understanding complex probability problems because it shows all the possible events we can analyze. For students, mastering the creation of power sets from sample spaces is essential to solve probability exercises accurately.

Probability for Managerial Life and Social Sciences

Probability plays a vital role in fields such as managerial life and social sciences, guiding decision-making processes that are often based on uncertain outcomes. This application of probability is used to evaluate the risk, predict future trends, and analyze data patterns to make informed decisions.
For instance, in business management, probability can help determine the likelihood of success for a new product, or the chances of a stock investment yielding positive returns. Similarly, in social sciences, researchers use probability to study patterns of social behavior and predict societal changes.
In such contexts, understanding events and sample spaces becomes incredibly practical. The concepts learned through exercises like the one provided can be employed to ascertain the likelihood of different scenarios, thereby allowing managers and scientists to assess options and make strategic choices based on quantified probabilities. To effectively communicate the relevance of probability to these scenarios, it's necessary to apply examples and context that relate directly to decision-making in managerial and social science issues.