In Exercises 49-53, use counting principles to find the probability.
52. A class of 40 students takes a statistics exam. The results are shown in the table at the left. Three students are selected at random. What is the probability that
b. all three students received a C or better?

Verified step by step guidance

1. Step 1: Identify the total number of students in the class. From the table, sum the number of students across all letter grades: A (8), B (10), C (12), D (6), and F (4). This gives a total of 40 students.
2. Step 2: Determine the number of students who received a grade of C or better. From the table, grades A, B, and C are considered 'C or better.' Add the number of students in these categories: A (8), B (10), and C (12). This gives a total of 30 students.
3. Step 3: Calculate the total number of ways to select 3 students from the class. Use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of students (40) and \( r \) is the number of students to be selected (3).
4. Step 4: Calculate the number of ways to select 3 students who received a grade of C or better. Again, use the combination formula \( \binom{n}{r} \), where \( n \) is the number of students who received C or better (30) and \( r \) is the number of students to be selected (3).
5. Step 5: Find the probability that all three selected students received a grade of C or better. Divide the number of favorable outcomes (from Step 4) by the total number of outcomes (from Step 3). The formula for probability is \( P = \frac{\text{favorable outcomes}}{\text{total outcomes}} \).

Key Concepts

Counting Principles

Counting principles, such as the fundamental counting principle, help determine the number of ways to choose or arrange items. In this context, it is essential for calculating the total number of combinations of students that can be selected from the class. Understanding how to apply these principles allows for accurate probability calculations based on the number of favorable outcomes versus total outcomes.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a ratio of favorable outcomes to total possible outcomes. In this scenario, the probability of selecting three students who received a C or better involves determining how many students fall into that category and how many ways they can be selected. This concept is fundamental for quantifying uncertainty in statistical analysis.

Combinations

Combinations refer to the selection of items from a larger set where the order does not matter. In this problem, we need to calculate the number of ways to choose three students from those who received a C or better. The combination formula, denoted as nCr, is crucial for determining how many different groups of students can be formed, which directly impacts the probability calculation.