Textbook Question
"In Exercises 19-22, determine whether the events are independent or dependent. Explain your reasoning.
21. Taking a driver's education course and passing the driver's license exam"
Ch. 3 - Probability
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 3, Problem 3.RE.44
In Exercises 41-44, perform the indicated calculation.
44. (5C3)/(10C3)
Verified step by step guidance1
Step 1: Understand the problem. The problem involves calculating a ratio of combinations. Specifically, you need to compute \( \frac{{5C3}}{{10C3}} \), where \( nCk \) represents the number of ways to choose \( k \) items from \( n \) items without regard to order.
Step 2: Recall the formula for combinations. The formula for \( nCk \) is \( \binom{n}{k} = \frac{{n!}}{{k!(n-k)!}} \), where \( n! \) is the factorial of \( n \).
Step 3: Compute \( 5C3 \) using the formula. Substitute \( n = 5 \) and \( k = 3 \) into the formula: \( \binom{5}{3} = \frac{{5!}}{{3!(5-3)!}} = \frac{{5!}}{{3! \cdot 2!}} \). Simplify this expression.
Step 4: Compute \( 10C3 \) using the formula. Substitute \( n = 10 \) and \( k = 3 \) into the formula: \( \binom{10}{3} = \frac{{10!}}{{3!(10-3)!}} = \frac{{10!}}{{3! \cdot 7!}} \). Simplify this expression.
Step 5: Divide the results of \( 5C3 \) and \( 10C3 \). Use the simplified values from Steps 3 and 4 to compute \( \frac{{5C3}}{{10C3}} \). Simplify the fraction to get the final result.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Combinations
Combinations refer to the selection of items from a larger set where the order does not matter. The notation 'nCr' represents the number of ways to choose 'r' items from 'n' items, calculated using the formula n! / (r!(n-r)!), where '!' denotes factorial. Understanding combinations is essential for solving problems involving group selections.
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Combinations
Factorial
A factorial, denoted as 'n!', is the product of all positive integers up to 'n'. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are fundamental in combinatorial calculations, as they help determine the total arrangements or selections of items in various scenarios.
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Probability
Probability is a measure of the likelihood of an event occurring, expressed as a ratio of favorable outcomes to the total number of possible outcomes. In the context of combinations, the ratio of two combinations, such as (5C3)/(10C3), can represent the probability of selecting a specific group from a larger set, providing insights into comparative likelihoods.
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Related Practice
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.
3. Experiment: Choosing a month of the year
Event: Choosing a month that begins with the letter J
Textbook Question
In Exercises 25 and 26, determine whether the events are mutually exclusive. Explain your reasoning.
25. Event A: Randomly select a red jelly bean from a jar.
Event B: Randomly select a yellow jelly bean from the jar.
Textbook Question
In Exercises 41-44, perform the indicated calculation.
42. 8P6
Textbook Question
Telephone Numbers The telephone numbers for a region of Pennsylvania have an area code of 570. The next seven digits represent the local telephone numbers for that region. These cannot begin with a 0 or 1. In Exercises 15 and 16, assume your cousin lives within the given area code.
16. What is the probability of not randomly generating your cousin's telephone number on the first try?
Textbook Question
In Exercises 19-22, determine whether the events are independent or dependent. Explain your reasoning.
19. Tossing a coin four times and getting four heads, and then tossing it a fifth time and getting a head