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
d. all three students received a B or a C?

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 grades: 8 (A) + 10 (B) + 12 (C) + 6 (D) + 4 (F) = 40 students.
2. Step 2: Determine the number of students who received a B or a C. From the table, the number of students with a B is 10, and the number of students with a C is 12. Therefore, the total number of students with a B or C is 10 + 12 = 22.
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 B or a C. Use the combination formula \( \binom{n}{r} \) again, where \( n \) is the number of students who received a B or C (22) and \( r \) is the number of students to be selected (3).
5. Step 5: Find the probability that all three selected students received a B or a C. 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 multiplication and addition rules, are fundamental in probability and statistics. They help determine the total number of possible outcomes in a given scenario. For example, when selecting students, the number of ways to choose a specific number of students from a larger group can be calculated using combinations, which is essential for finding probabilities.

Combinations

Combinations refer to the selection of items from a larger set where the order does not matter. In this context, when selecting three students from the class, we use combinations to calculate how many different groups of three can be formed from those who received a B or C. The formula for combinations is n! / [r!(n-r)!], where n is the total number of items, and r is the number of items to choose.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. To find the probability that all three selected students received a B or C, we calculate the number of favorable outcomes (selecting three students from the B and C groups) divided by the total number of possible outcomes (selecting any three students from the entire class). This ratio provides the required probability.