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Ch. 4 - Probability
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 4 - ProbabilityProblem 4.c.4d
Chapter 4, Problem 4.c.4d

Sampling Eye Color Based on a study by Dr. P. Sorita Soni at Indiana University, assume that eye colors in the United States are distributed as follows: 40% brown, 35% blue, 12% green, 7% gray, 6% hazel.


d. If two people are randomly selected, what is the probability that at least one of them has brown eyes?

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Step 1: Understand the problem. We are tasked with finding the probability that at least one of two randomly selected people has brown eyes. This is a complementary probability problem, where we can use the complement rule: P(at least one has brown eyes) = 1 - P(neither has brown eyes).
Step 2: Calculate the probability that a single person does NOT have brown eyes. Since the probability of having brown eyes is 40% (or 0.4), the probability of NOT having brown eyes is 1 - 0.4 = 0.6.
Step 3: Calculate the probability that neither of the two people has brown eyes. Since the two selections are independent, the probability that both do not have brown eyes is the product of their individual probabilities: P(neither has brown eyes) = P(not brown) × P(not brown) = 0.6 × 0.6.
Step 4: Use the complement rule to find the probability that at least one of the two people has brown eyes. Subtract the probability of neither having brown eyes from 1: P(at least one has brown eyes) = 1 - P(neither has brown eyes).
Step 5: Substitute the value from Step 3 into the formula from Step 4 to complete the calculation. This will give you the final probability that at least one of the two people has brown eyes.

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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 helps determine the chance of selecting individuals with specific eye colors from a population. Understanding basic probability rules, such as the complement rule, is essential for solving problems involving multiple selections.
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Introduction to Probability

Complement Rule

The complement rule states that the probability of an event occurring is equal to 1 minus the probability of it not occurring. In this case, to find the probability that at least one of the two selected individuals has brown eyes, it is often easier to first calculate the probability that neither has brown eyes and then subtract that from 1.
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Complementary Events

Independent Events

Independent events are those whose outcomes do not affect each other. When selecting two people randomly, the probability of one person's eye color does not influence the other's. This concept is crucial for calculating the combined probabilities of multiple selections, as it allows for the multiplication of individual probabilities.
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Probability of Multiple Independent Events
Related Practice
Textbook Question

Florida Pick 3 In the Florida Pick 3 lottery, you can place a “straight” bet of \(1 by selecting the exact order of three digits between 0 and 9 inclusive (with repetition allowed), so the probability of winning is 1/1000. If the same three numbers are drawn in the same order, you collect \)500, so your net profit is \$499.


c. Is there much of a difference between the actual odds against winning and the payoff odds?

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Textbook Question

Mega Millions As of this writing, the Mega Millions lottery is run in 44 states. Winning the jackpot requires that you select the correct five different numbers from 1 to 70 and, in a separate drawing, you must also select the correct single number from 1 to 25.


c. How does the probability compare to the probability for the old Mega Millions game which involved the selection of five different numbers between 1 and 75 and a separate single number between 1 and 15?

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Textbook Question

Births in the United States In the United States, the true probability of a baby being a boy is 0.512 (based on the data available at this writing). For a family having three children, find the following.


d. The probability that at least one of the children is a girl.

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Textbook Question

Subjective Probability Estimate the probability that the next time you watch a TV news report, it includes a story about a plane crash.

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Textbook Question

Phase I of a Clinical Trial A clinical test on humans of a new drug is normally done in three phases. Phase I is conducted with a relatively small number of healthy volunteers. For example, a phase I test of bexarotene involved only 14 subjects. Assume that we want to treat 14 healthy humans with this new drug and we have 16 suitable volunteers available.


c. If 14 subjects are randomly selected and treated at the same time, what is the probability of selecting the 14 youngest subjects?

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Textbook Question

In Exercises 6–10, use the following results from tests of an experiment to test the effectiveness of an experimental vaccine for children (based on data from USA Today). Express all probabilities in decimal form.



If 1 of the 1602 subjects is randomly selected, find the probability of getting 1 who had the vaccine treatment and developed flu.

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