Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

An experiment consists of selecting a letter at random from the letters in the word MASSACHUSETTS and observing the outcomes. a. What is an appropriate sample space for this experiment? b. Describe the event "the letter selected is a vowel."

Step-by-Step Solution

Answer

a. The appropriate sample space for this experiment is \(S = \{ M, A, S, C, H, U, T, E \}\). b. The event "the letter selected is a vowel" is represented as \(V = \{ A, E, U \}\).

Step 1: Identify all distinct letters in 'MASSACHUSETTS'

To find the sample space, we need to list all the possible outcomes that can be obtained when selecting a letter randomly from the word "MASSACHUSETTS." The distinct letters are: M, A, S, C, H, U, T, and E.

Step 2: Write down the sample space

Now that we have identified all distinct letters in the word "MASSACHUSETTS." Let's represent the sample space as a set, with each element corresponding to a distinct letter. The sample space is: \(S = \{ M, A, S, C, H, U, T, E \}\). #b. Describe the event "the letter selected is a vowel."#

Step 1: Identify the vowels in 'MASSACHUSETTS'

To describe the event of selecting a vowel, we need to list all the vowels in the word "MASSACHUSETTS". The vowels are A, E, and U.

Step 2: Write down the event

Now that we have identified all vowels in the word "MASSACHUSETTS," let's represent the event as a set, with each element corresponding to a vowel. The event "the letter selected is a vowel" is: \(V = \{ A, E, U \}\).

Key Concepts

Probability Theory

Probability theory is a branch of mathematics concerned with the analysis of random phenomena. It involves the study of uncertainty and how to quantify it through the probabilities of different outcomes. At its root, probability theory allows us to make calculated decisions based on incomplete information, a skill that's widely applicable in fields such as finance, science, engineering, and everyday life.

For example, in the problem provided, the task to determine the sample space for randomly selecting a letter from 'MASSACHUSETTS' draws directly from probability theory. By listing all the distinct letters in the word, one can set up the foundation for calculating probabilities of various events, such as choosing a particular letter. Probability theory guides us to systematically analyze the potential outcomes and calculate their likelihood.

Random Experiment Outcomes

The outcomes of a random experiment are the possible results that can occur. In probability, we often group these outcomes into a set known as the sample space. It is vital to ensure that the sample space is well-defined and comprehensive, covering all possible outcomes without any omissions or duplications.

In our specific exercise, the sample space comprises the distinct letters found within 'MASSACHUSETTS'. By recognizing the unique letters, we prevent overcounting certain outcomes, which could skew the probabilities. Understanding and accurately listing outcomes is crucial because they serve as the basis for all subsequent probability calculations, ensuring they reflect the true nature of the random experiment.

Describing Events Probability

Describing events in probability refers to specifying a subset of outcomes from the sample space that share a common property for which we wish to calculate the probability. In the context of our exercise, the event is selecting a vowel from 'MASSACHUSETTS'. The probability of this event is quantified by comparing the number of favorable outcomes (in this case, vowels) to the total number of outcomes within the sample space.

To achieve this, we first identify the common characteristic (being a vowel), list the letters that possess this property, and then describe the event as the set of those letters, such as \(V = \{ A, E, U \}\). Understanding how to describe such events is a key step towards calculating probabilities, as it allows for a clear definition of what we are measuring and thus helps to determine the likelihood in a precise and mathematical manner.