Problem 14 - Precalculus with Limits

Question

In Exercises 9 - 14, determine the sample space for the experiment. A sales representative makes presentations about a product in three homes per day. In each home, there may be a sale (denote by \( S \)) or there may be no sale(denote by \( F \)).

Step-by-Step Solution

Answer

The sample space for the experiment is: \( \{ SSS, SSF, SFS, SFF, FSS, FSF, FFS, FFF \} \).

Step 1: Understand the Possible Outcomes

In each home, the sales representative can either make a sale or not. Hence, there are two possible outcomes for each presentation: a sale (\( S \)) or no sale (\( F \)).

Step 2: Determine the Sample Space for One Home

For a single home the possible outcomes are either \( S \) or \( F \). These can essentially be seen as the sample points for one presentation.

Step 3: Expand the Sample Space for Three Homes

Now, we need to account for three homes, so we'll have three lots of these possible outcomes. Remember, the outcomes for each home are independent of each other. With three events, each having two possibilities, the total number of outcomes will be \(2^3 = 8\). These outcomes are: \( SSS, SSF, SFS, SFF, FSS, FSF, FFS, FFF \).

Key Concepts

Probability Theory

Probability theory is the branch of mathematics that deals with quantifying the likelihood of events occurring. It has become a fundamental tool in many fields, including statistics, economics, and science. In the context of the sales representative example, probability theory would help us determine the likelihood of various sales outcomes over the course of three home presentations.

To calculate the probability of an event, we first need to understand the sample space, which is the set of all possible outcomes. For one presentation, our sample space is {S, F}. Since each presentation is independent, we can calculate the probabilities of combinations of outcomes using the rules of probability. For instance, if the probability of making a sale at one home is 0.5 and not making a sale is also 0.5, the probability of three sales in a row would be calculated by multiplying these individual probabilities.

Combinatorics

Combinatorics is the area of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It helps us to understand and quantify the number of ways certain events can occur.

In our sales scenario, combinatorics enables us to calculate the different combinations in which sales can occur across three homes. Each home represents an independent event with two outcomes. The fundamental principle of counting tells us that if one event can occur in 'm' ways and another independent event can occur in 'n' ways, then the two events together can occur in 'm * n' ways.

Applying the Counting Principle

Since there are 2 ways the outcome of a presentation can go (sale or no sale) and there are 3 presentations, we use the principle to calculate the total possible outcomes as 2 * 2 * 2, which equals 8 distinct outcomes. These different outcomes, such as 'SSF' or 'FFS', represent the sample space for the experiment.

Independent Events

Independent events are outcomes where the occurrence of one event does not influence the probability of another. This is a crucial concept in probability theory because it simplifies the calculation of probabilities for multiple events.

In the case of the sales representative, the outcome of each home presentation is independent from the others. This means the result of the sale in the first home does not affect the potential sale in the second or third home.

Understanding Independence

With a concrete grasp of independent events, one can understand that the likelihood of a sale at any given home is unaffected by previous results. Therefore, when calculating the probability of various combinations of sales and no-sales across the three homes, each home's result is considered in isolation. This principle was used in expanding the sample space for the three homes in the original exercise.