Suppose a jar contains 6 blue, 3 green, and 2 red marbles. What is the probability of randomly selecting three marbles at the same time and all of them being blue?
Suppose a jar contains 6 blue, 3 green, and 2 red marbles. What is the probability of randomly selecting three marbles at the same time and all of them being blue?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 34T
Related questions
Question
![**Problem Statement:**
Suppose a jar contains 6 blue, 3 green, and 2 red marbles. What is the probability of randomly selecting three marbles at the same time and all of them being blue?
**Explanation:**
To solve this problem, we need to calculate the probability of drawing 3 blue marbles from a total of 11 marbles (6 blue, 3 green, 2 red). This involves using combinations to determine the number of favorable outcomes and the total possible outcomes.
**Step 1: Calculate the Total Number of Ways to Select 3 Marbles**
The total number of ways to select 3 marbles from 11 is given by the combination formula:
\[ C(n, k) = \frac{n!}{k!(n-k)!} \]
where \( n \) is the total number of marbles, and \( k \) is the number of marbles to choose.
\[ C(11, 3) = \frac{11!}{3!(11-3)!} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165 \]
**Step 2: Calculate the Number of Ways to Select 3 Blue Marbles**
The number of ways to select 3 blue marbles from the 6 available is:
\[ C(6, 3) = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
**Step 3: Calculate the Probability**
The probability of drawing 3 blue marbles is the ratio of successful outcomes to total outcomes:
\[ \text{Probability} = \frac{\text{Number of ways to select 3 blue marbles}}{\text{Total number of ways to select 3 marbles}} = \frac{20}{165} \approx 0.1212 \]
Therefore, the probability is approximately 0.1212, or 12.12%.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7e488f-dfc3-4cc0-b337-e209f551039c%2F88298757-3894-4bde-b172-cad69df9b135%2Fu55buqm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Suppose a jar contains 6 blue, 3 green, and 2 red marbles. What is the probability of randomly selecting three marbles at the same time and all of them being blue?
**Explanation:**
To solve this problem, we need to calculate the probability of drawing 3 blue marbles from a total of 11 marbles (6 blue, 3 green, 2 red). This involves using combinations to determine the number of favorable outcomes and the total possible outcomes.
**Step 1: Calculate the Total Number of Ways to Select 3 Marbles**
The total number of ways to select 3 marbles from 11 is given by the combination formula:
\[ C(n, k) = \frac{n!}{k!(n-k)!} \]
where \( n \) is the total number of marbles, and \( k \) is the number of marbles to choose.
\[ C(11, 3) = \frac{11!}{3!(11-3)!} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165 \]
**Step 2: Calculate the Number of Ways to Select 3 Blue Marbles**
The number of ways to select 3 blue marbles from the 6 available is:
\[ C(6, 3) = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
**Step 3: Calculate the Probability**
The probability of drawing 3 blue marbles is the ratio of successful outcomes to total outcomes:
\[ \text{Probability} = \frac{\text{Number of ways to select 3 blue marbles}}{\text{Total number of ways to select 3 marbles}} = \frac{20}{165} \approx 0.1212 \]
Therefore, the probability is approximately 0.1212, or 12.12%.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning