Textbook Question
Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
20. Coin and Die A coin is tossed and a die is rolled. Find the probability of tossing a tail and then rolling a number greater than 2.
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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 3.4.40
Board of Directors The University of Colorado Board of Directors has 23 members. One member serves as board chair and another serves as vice chair. Given the names of the 23
board members, what is the probability of randomly selecting the name of the chair and the name of the vice chair? (Source: University of Colorado)
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with finding the probability of randomly selecting one person as the chair and another as the vice chair from a group of 23 members. This is a problem involving combinations and probabilities.
Step 2: Calculate the total number of ways to select the chair and vice chair. Since the selection is ordered (chair first, then vice chair), this is a permutation problem. The number of permutations is given by the formula: P(n, r) = n! / (n - r)!, where n is the total number of members (23) and r is the number of positions to fill (2).
Step 3: Substitute the values into the permutation formula. Using MathML, the formula becomes: . Here, n = 23 and r = 2, so the calculation is: .
Step 4: Simplify the expression. The factorials simplify to: , because the remaining terms cancel out when dividing by .
Step 5: Calculate the probability. Since there is only one specific pair of names that can be selected as the chair and vice chair, the probability is the reciprocal of the total number of permutations. This is given by: .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability
Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it refers to the chance of randomly selecting specific individuals from a group. The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
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Combinations
Combinations refer to the selection of items from a larger set where the order does not matter. In the context of this question, it involves selecting the chair and vice chair from the board members. Understanding combinations is essential for calculating probabilities when the order of selection is irrelevant.
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05:22Combinations
Random Selection
Random selection is a process where each member of a group has an equal chance of being chosen. This concept is crucial for ensuring that the probability calculations are valid, as it assumes that the selection of the chair and vice chair is unbiased and purely based on chance, without any influence from external factors.
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07:09Intro to Random Variables & Probability Distributions
Related Practice
Textbook Question
"Classifying Events as Independent or Dependent In Exercises 9-14, determine whether the events are independent or dependent. Explain your reasoning.
9. Selecting a king from a standard deck of 52 playing cards, replacing it, and then selecting a queen from the deck"
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Textbook Question
Writing In Exercises 89 and 90, write a statement that represents the complement of the probability.
90. The probability of randomly choosing a car with more than one cause for showing its "CHECK ENGINE" light from the population of vehicles showing "CHECK ENGINE" lights.
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Textbook Question
Classifying Events as Independent or Dependent In Exercises 9-14, determine whether the events are independent or dependent. Explain your reasoning.
14. A ball is selected from a bin of balls numbered from 1 through 52. It is replaced, and then a second numbered ball is selected from the bin.
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Textbook Question
37. Water Pollution An environmental agency is analyzing water samples from 80 lakes for pollution. Five of the lakes have dangerously high levels of dioxin. Six lakes are randomly selected from the sample. Use technology to find how many ways one polluted lake and five nonpolluted lakes can be chosen.
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Textbook Question
Using a Frequency Distribution to Find Probabilities In Exercises 49-52, use the frequency distribution at the left, which shows the population of the United States by age group, to find the probability that a U.S. resident chosen at random is in the age range. (Source: U.S. Census Bureau)
52. 65 years old and older
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