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Ch. 3 - Probability
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 3 - ProbabilityProblem 3.R.52d
Chapter 3, Problem 3.R.52d

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?
Table showing letter grades and the number of students who received each grade: A-8, B-10, C-12, D-6, F-4.

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.
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.
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).
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).
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 \( P = \frac{\text{favorable outcomes}}{\text{total outcomes}} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Fundamental Counting Principle

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.
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Combinations

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.
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Introduction to Probability
Related Practice
Textbook Question

In Exercises 17 and 18, use the table, which shows the numbers of first-time and repeat U.S. nursing students taking the National Council Licensure Examination (NCLEX-RN® exam) to pass or fail in a recent year. (Adapted from National Council Licensure Examinations)

17. Find the probability that a student took the exam for the first time, given that the student failed.

Textbook Question

In Exercises 45-48, use combinations and permutations.

46. Five players on a basketball team must each choose one of the five players on the opposing team to defend. In how many ways can the players choose their defensive assignments?

Textbook Question

"In Exercises 1-4, identify the sample space of the probability experiment and determine the number of outcomes in the event. Draw a tree diagram when appropriate.

1. Experiment: Tossing four coins

Event: Getting three heads"

Textbook Question

In Exercises 45-48, use combinations and permutations.

48. An employer must hire 2 people from a list of 13 applicants. In how many ways can the employer choose to hire the two people?

Textbook Question

In Exercises 49-53, use counting principles to find the probability.

50. A security code consists of three letters and one digit. The first letter cannot be A, B, or C. What is the probability of guessing the security code on the first try?

Textbook Question

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?