Problem 11 - Precalculus with Limits

Question

In Exercises 9 - 14, determine the sample space for the experiment. A taste tester has to rank three varieties of yogurt, \( A \), \( B \),and \( C \), according to preference.

Step-by-Step Solution

Answer

The sample space for the experiment is \( S = \{ABC, ACB, BAC, BCA, CAB, CBA\} \). There are 6 different arrangements possible.

Step 1: Setup the problem

We need to find different arrangements for the three types of yogurt, so we approach this as a permutation problem. A permutation is an arrangement of elements where order is important.

Step 2: Permutation Calculation

The number of ways to rank three objects is given by the formula \( P(n) = n! \), where \( n! \) denotes the factorial of \( n \). Factorial is a function that multiplies \( n \) by every natural number less than \( n \). In our case, \( n = 3 \) because we have three different types of yogurt.

Step 3: Calculate Factorial

Using the formula and given \( n = 3 \), we calculate \( P(3) = 3! = 3*2*1 = 6 \). So there are 6 different ways the taste tester can rank the three types of yogurt.

Step 4: List All Possible Arrangements

Finally, let's list out all the possible arrangements: \[ ABC, ACB, BAC, BCA, CAB, CBA \] Each of these represents a different ranking of the three types of yogurt.

Key Concepts

Understanding Sample Space

When diving into the world of probability, the concept of a 'sample space' is a fundamental starting point. It refers to the complete set of all possible outcomes of a particular experiment or situation. Imagine it as a collection capturing every potential result that could occur.

In the context of our yogurt taste-testing scenario, the sample space consists of all possible rankings the tester could give to the three yogurt varieties labeled A, B, and C. Understanding the sample space is crucial as it forms the basis for calculating probabilities; any event we are interested in is a subset of this larger set.

To help students visualize this concept, one could imagine laying out all possible options on the table, much like spreading out a deck of cards to see every individual card at once. By grasping the sample space, students gain a clear picture of the experiment’s scope and are better equipped to tackle more complex probability questions.

The Role of Factorial in Permutations

The concept of a factorial, denoted by an exclamation mark (!), plays a vital role in counting permutations. A factorial represents the product of all positive integers up to a given number. Specifically, for any non-negative integer n, the factorial of n, or \( n! \), is the product of all positive integers less than or equal to n. It's defined as \( n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1 \).

For instance, \( 3! = 3 \times 2 \times 1 = 6 \). Factorials grow extremely quickly with larger numbers, making them powerful tools in computing permutations where order matters. Students can better understand factorials by working through a few practice exercises where they multiply consecutive descending numbers, starting from their chosen n.

Permutation and Arrangements

Permutation is a cornerstone concept when it comes to arrangements, especially in scenarios where the order is important. It tells us the number of ways we can organize a set of items. To calculate the permutations of n distinct items, we use the factorial of n, symbolized as \( n! \).

In our yogurt ranking illustration, we're dealing with 3 distinct flavors, A, B, and C. Since order matters (a ranking of ABC is different from ACB), we are looking for the number of permutations of these 3 items: \( P(3) = 3! \). This equation yields 6 distinct arrangements, hence giving us the full sample space of possible rankings.

For educational purposes, working through the process of listing each arrangement can reinforce the understanding of permutations. This hands-on activity helps create a tangible link between the mathematical concept and its practical application, which is essential for grasping the abstract notion of permutations.