Textbook Question
High Fives
b. If n mathletes shake hands with each other exactly once, what is the total number of handshakes?
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Ch. 4 - Probability
Triola14th EditionElementary StatisticsISBN: 9780137366446
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Table of contents
- Ch. 1 - Introduction to Statistics88
- Ch. 2 - Exploring Data with Tables and Graphs70
- Ch. 3 - Describing, Exploring, and Comparing Data66
- Ch. 4 - Probability112
- Ch. 5 - Discrete Probability Distributions109
- Ch. 6 - Normal Probability Distributions122
- Ch. 7 - Estimating Parameters and Determining Sample Sizes107
- Ch. 8 - Hypothesis Testing75
- Ch. 9 - Inferences from Two Samples94
- Ch. 10 - Correlation and Regression80
- Ch. 11 - Goodness-of-Fit and Contingency Tables29
- Ch. 12 - Analysis of Variance42
Chapter 4, Problem 4.2.26b
Alarm Clock Life Hack Each of us must sometimes wake up early for something really important, such as a final exam, job interview, or an early flight. (Professional golfer Jim Furyk was disqualified from a tournament when his cellphone lost power and he overslept.) Assume that a battery-powered alarm clock has a 0.005 probability of failure, a smartphone alarm clock has a 0.052 probability of failure, and an electric alarm clock has a 0.001 probability of failure.
b. If you use a battery-powered alarm clock and a smartphone alarm clock, what is the probability that they both fail? What is the probability that both of them do not fail?
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with finding two probabilities: (1) the probability that both a battery-powered alarm clock and a smartphone alarm clock fail, and (2) the probability that neither of them fails. The failure probabilities are given as 0.005 for the battery-powered alarm clock and 0.052 for the smartphone alarm clock.
Step 2: To calculate the probability that both fail, use the multiplication rule for independent events. The formula is P(A and B) = P(A) * P(B), where A is the event that the battery-powered alarm clock fails, and B is the event that the smartphone alarm clock fails. Substitute the given probabilities into the formula: P(both fail) = 0.005 * 0.052.
Step 3: To calculate the probability that neither fails, first find the probability that each clock does not fail. For the battery-powered alarm clock, the probability of not failing is 1 - 0.005 = 0.995. For the smartphone alarm clock, the probability of not failing is 1 - 0.052 = 0.948.
Step 4: Use the multiplication rule for independent events again to find the probability that neither fails. The formula is P(neither fails) = P(not A) * P(not B), where not A and not B are the events that the battery-powered alarm clock and the smartphone alarm clock do not fail, respectively. Substitute the probabilities: P(neither fails) = 0.995 * 0.948.
Step 5: Summarize the results. You now have the formulas to calculate both probabilities: (1) P(both fail) = 0.005 * 0.052, and (2) P(neither fails) = 0.995 * 0.948. Perform the calculations to find the final numerical values if needed.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability of Independent Events
In probability theory, independent events are those whose outcomes do not affect each other. For example, the failure of a battery-powered alarm clock does not influence the failure of a smartphone alarm clock. To find the probability of both events occurring, you multiply their individual probabilities. This concept is crucial for calculating the likelihood of multiple alarms failing simultaneously.
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Probability of Multiple Independent Events
Complementary Probability
Complementary probability refers to the likelihood of an event not occurring. If the probability of an event happening is P, then the probability of it not happening is 1 - P. This concept is essential for determining the probability that both alarm clocks do not fail, as it allows us to calculate the complement of their failure probabilities.
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4:23Complementary Events
Joint Probability
Joint probability is the probability of two events occurring at the same time. In this scenario, we are interested in the joint probability of both the battery-powered and smartphone alarm clocks failing. This is calculated by multiplying the individual probabilities of failure for each clock, which provides insight into the overall risk of relying on both alarms.
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5:37Introduction to Probability
Related Practice
Textbook Question
In Exercises 21-28, find the probability and answer the questions.
Genetics: Eye Color Each of two parents has the genotype brown/blue, which consists of the pair of alleles that determine eye color, and each parent contributes one of those alleles to a child. Assume that if the child has at least one brown allele, that color will dominate and the eyes will be brown. (The actual determination of eye color is more complicated than that.)
b. What is the probability that a child of these parents will have the blue/blue genotype?
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Textbook Question
In Exercises 21-28, find the probability and answer the questions.
Guessing Birthdays On their first date, Kelly asks Mike to guess the date of her birth, not including the year.
b. Would it be unlikely for him to guess correctly on his first try?
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Textbook Question
In Exercises 9–20, use the data in the following table, which lists survey results from high school drivers at least 16 years of age (based on data from “Texting While Driving and Other Risky Motor Vehicle Behaviors Among U.S. High School Students,” by O’Malley, Shults, and Eaton, Pediatrics, Vol. 131, No. 6). Assume that subjects are randomly selected from those included in the table. Hint: Be very careful to read the question correctly.
Drinking and Driving If two of the high school drivers are randomly selected, find the probability that they both drove when drinking alcohol.
b. Assume that the selections are made without replacement. Are the events independent?
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Textbook Question
In Exercises 29 and 30, find the probabilities and indicate when the “5% guideline for cumbersome calculations” is used.
Medical Helicopters In a study of helicopter usage and patient survival, results were obtained from 47,637 patients transported by helicopter and 111,874 patients transported by ground (based on data from “Association Between Helicopter vs Ground Emergency Medical Services and Survival for Adults with Major Trauma,” by Galvagno et al., Journal of the American Medical Association, Vol. 307, No. 15).
b. If 5 of the subjects in the study are randomly selected without replacement, what is the probability that all of them were transported by helicopter?
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Textbook Question
In Exercises 21–24, use these results from the “1-Panel-THC” test for marijuana use, which is provided by the company Drug Test Success: Among 143 subjects with positive test results, there are 24 false positive (incorrect) results; among 157 negative results, there are 3 false negative (incorrect) results. (Hint: Construct a table similar to Table 4-1.)
Testing for Marijuana Use
b. How many of the subjects had a true negative result?
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