Nico Owns 11 Instructional Piano Books. Two Are Beginner Books, Six Are Intermediate Books, And Three Are Advanced Books.If Two Books Are Randomly Chosen From The Collection, One At A Time, And Replaced After Each Pick, What Is The Probability That He
Introduction
In this article, we will explore the concept of probability and its application in a real-world scenario. Nico, a piano enthusiast, owns a collection of 11 instructional piano books, consisting of beginner, intermediate, and advanced levels. We will calculate the probability of choosing two books randomly from the collection, one at a time, and replaced after each pick.
Understanding Probability
Probability is a measure of the likelihood of an event occurring. It is usually expressed as a value between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. In this case, we want to find the probability of choosing two books that meet a certain condition.
Calculating Probability
To calculate the probability of choosing two books, we need to consider the total number of possible outcomes and the number of favorable outcomes. Since the books are replaced after each pick, the probability remains the same for each pick.
Let's break down the problem into smaller parts:
- Total number of books: 11
- Number of beginner books: 2
- Number of intermediate books: 6
- Number of advanced books: 3
We want to find the probability of choosing two books that meet a certain condition. Let's consider the following scenarios:
Scenario 1: Choosing two beginner books
- Total number of possible outcomes: 11 (since there are 11 books in total)
- Number of favorable outcomes: 2 (since there are 2 beginner books)
- Probability of choosing a beginner book on the first pick: 2/11
- Probability of choosing a beginner book on the second pick: 2/11 (since the books are replaced after each pick)
- Probability of choosing two beginner books: (2/11) × (2/11) = 4/121
Scenario 2: Choosing two intermediate books
- Total number of possible outcomes: 11
- Number of favorable outcomes: 6
- Probability of choosing an intermediate book on the first pick: 6/11
- Probability of choosing an intermediate book on the second pick: 6/11
- Probability of choosing two intermediate books: (6/11) × (6/11) = 36/121
Scenario 3: Choosing two advanced books
- Total number of possible outcomes: 11
- Number of favorable outcomes: 3
- Probability of choosing an advanced book on the first pick: 3/11
- Probability of choosing an advanced book on the second pick: 3/11
- Probability of choosing two advanced books: (3/11) × (3/11) = 9/121
Scenario 4: Choosing one beginner book and one intermediate book
- Total number of possible outcomes: 11
- Number of favorable outcomes: 2 × 6 = 12
- Probability of choosing a beginner book on the first pick: 2/11
- Probability of choosing an intermediate book on the second pick: 6/11
- Probability of choosing an intermediate book on the first pick: 6/11
- Probability of choosing a beginner book on the second pick: 2/11
- Probability of choosing one beginner book and one intermediate book: (2/11) × (6/11) + (6/11) × (2/11) = 24/121
Scenario 5: Choosing one beginner book and one advanced book
- Total number of possible outcomes: 11
- Number of favorable outcomes: 2 × 3 = 6
- Probability of choosing a beginner book on the first pick: 2/11
- Probability of choosing an advanced book on the second pick: 3/11
- Probability of choosing an advanced book on the first pick: 3/11
- Probability of choosing a beginner book on the second pick: 2/11
- Probability of choosing one beginner book and one advanced book: (2/11) × (3/11) + (3/11) × (2/11) = 12/121
Scenario 6: Choosing one intermediate book and one advanced book
- Total number of possible outcomes: 11
- Number of favorable outcomes: 6 × 3 = 18
- Probability of choosing an intermediate book on the first pick: 6/11
- Probability of choosing an advanced book on the second pick: 3/11
- Probability of choosing an advanced book on the first pick: 3/11
- Probability of choosing an intermediate book on the second pick: 6/11
- Probability of choosing one intermediate book and one advanced book: (6/11) × (3/11) + (3/11) × (6/11) = 36/121
Conclusion
In conclusion, the probability of choosing two books that meet a certain condition depends on the specific condition. We have calculated the probability of choosing two beginner books, two intermediate books, two advanced books, one beginner book and one intermediate book, one beginner book and one advanced book, and one intermediate book and one advanced book.
The probability of choosing two beginner books is 4/121, the probability of choosing two intermediate books is 36/121, the probability of choosing two advanced books is 9/121, the probability of choosing one beginner book and one intermediate book is 24/121, the probability of choosing one beginner book and one advanced book is 12/121, and the probability of choosing one intermediate book and one advanced book is 36/121.
Final Answer
The final answer is not a single number, but rather a set of probabilities for different scenarios. However, if we want to find the overall probability of choosing two books that meet a certain condition, we need to consider all the possible scenarios and their corresponding probabilities.
Let's assume that the condition is to choose two books that are either beginner or intermediate. In this case, we can add the probabilities of choosing two beginner books, two intermediate books, and one beginner book and one intermediate book.
The probability of choosing two beginner books is 4/121, the probability of choosing two intermediate books is 36/121, and the probability of choosing one beginner book and one intermediate book is 24/121. Therefore, the overall probability of choosing two books that are either beginner or intermediate is:
(4/121) + (36/121) + (24/121) = 64/121
This means that the probability of choosing two books that are either beginner or intermediate is 64/121, or approximately 0.53.
Note
Q: What is the probability of choosing two beginner books from Nico's collection?
A: The probability of choosing two beginner books from Nico's collection is 4/121, or approximately 0.033.
Q: What is the probability of choosing two intermediate books from Nico's collection?
A: The probability of choosing two intermediate books from Nico's collection is 36/121, or approximately 0.298.
Q: What is the probability of choosing two advanced books from Nico's collection?
A: The probability of choosing two advanced books from Nico's collection is 9/121, or approximately 0.075.
Q: What is the probability of choosing one beginner book and one intermediate book from Nico's collection?
A: The probability of choosing one beginner book and one intermediate book from Nico's collection is 24/121, or approximately 0.198.
Q: What is the probability of choosing one beginner book and one advanced book from Nico's collection?
A: The probability of choosing one beginner book and one advanced book from Nico's collection is 12/121, or approximately 0.099.
Q: What is the probability of choosing one intermediate book and one advanced book from Nico's collection?
A: The probability of choosing one intermediate book and one advanced book from Nico's collection is 36/121, or approximately 0.298.
Q: What is the overall probability of choosing two books that are either beginner or intermediate from Nico's collection?
A: The overall probability of choosing two books that are either beginner or intermediate from Nico's collection is 64/121, or approximately 0.53.
Q: How many books are in Nico's collection?
A: There are 11 books in Nico's collection.
Q: How many beginner books are in Nico's collection?
A: There are 2 beginner books in Nico's collection.
Q: How many intermediate books are in Nico's collection?
A: There are 6 intermediate books in Nico's collection.
Q: How many advanced books are in Nico's collection?
A: There are 3 advanced books in Nico's collection.
Q: Are the books in Nico's collection replaced after each pick?
A: Yes, the books in Nico's collection are replaced after each pick.
Q: What is the probability of choosing a beginner book on the first pick?
A: The probability of choosing a beginner book on the first pick is 2/11, or approximately 0.182.
Q: What is the probability of choosing an intermediate book on the first pick?
A: The probability of choosing an intermediate book on the first pick is 6/11, or approximately 0.545.
Q: What is the probability of choosing an advanced book on the first pick?
A: The probability of choosing an advanced book on the first pick is 3/11, or approximately 0.273.
Q: What is the probability of choosing a beginner book on the second pick?
A: The probability of choosing a beginner book on the second pick is 2/11, or approximately 0.182.
Q: What is the probability of choosing an intermediate book on the second pick?
A: The probability of choosing an intermediate book on the second pick is 6/11, or approximately 0.545.
Q: What is the probability of choosing an advanced book on the second pick?
A: The probability of choosing an advanced book on the second pick is 3/11, or approximately 0.273.
Q: Can you explain the concept of probability in simple terms?
A: Probability is a measure of the likelihood of an event occurring. It is usually expressed as a value between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. In this case, we are calculating the probability of choosing two books from Nico's collection that meet a certain condition.