Ch. 3 - Probability

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 3 - Probability

Problem 3.2.27c

Chapter 3, Problem 3.2.27c

Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
27. Blood Types The probability that a person of Asian descent in the United States has type O+ blood is 39%. At random, six people of Asian descent in the United States are selected. (Source: American National Red Cross)
c. Find the probability that at least one of the six has type O+ blood.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the probability that at least one of the six randomly selected people of Asian descent has type O+ blood. This is a complementary probability problem, where we will first calculate the probability that none of the six people have type O+ blood and then subtract this value from 1.

Step 2: Define the probability of success and failure. The probability that a person has type O+ blood is 0.39 (success), and the probability that a person does not have type O+ blood is 1 - 0.39 = 0.61 (failure).

Step 3: Use the Multiplication Rule to calculate the probability that none of the six people have type O+ blood. Since the events are independent, the probability that all six people do not have type O+ blood is given by \( P(\text{none}) = (0.61)^6 \).

Step 4: Use the complement rule to find the probability that at least one of the six people has type O+ blood. The complement rule states that \( P(\text{at least one}) = 1 - P(\text{none}) \). Substitute the value of \( P(\text{none}) \) from Step 3 into this formula.

Step 5: Simplify the expression \( P(\text{at least one}) = 1 - (0.61)^6 \) to find the final probability. This will give the probability that at least one of the six people has type O+ blood.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Multiplication Rule of Probability

The Multiplication Rule states that the probability of two independent events occurring together is the product of their individual probabilities. In this context, if the probability of one person having type O+ blood is 39%, the probability of not having type O+ blood is 61%. This rule is essential for calculating the probability of multiple independent selections.

Complementary Probability

Complementary probability refers to the likelihood of an event not occurring. For instance, if the probability of a person having type O+ blood is 39%, the complementary probability of not having type O+ blood is 61%. This concept is crucial for solving the problem, as it allows us to find the probability of at least one person having type O+ blood by first calculating the probability that none do.

Binomial Probability

Binomial probability deals with scenarios where there are a fixed number of independent trials, each with two possible outcomes (success or failure). In this case, selecting six people can be modeled as a binomial experiment where 'success' is defined as having type O+ blood. Understanding this concept helps in calculating the overall probability of at least one success in multiple trials.

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Related Practice

Textbook Question

26. Eye Survey The table shows the results of a survey that asked 3203 people whether they wore contacts or glasses. A person is selected at random from the sample. Find the probability of each event.

d. The person is male or does not wear glasses.

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Textbook Question

Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.

26. Worst President In a sample of 1500 adult U.S. citizens, 690 said that Donald Trump was the worst president in U.S. history. Three adult U.S. citizens are selected at random.

(Adapted from YouGov)

c. Find the probability that at most two of the three adult U.S. citizens say that Donald Trump was the worst president in U.S. history.

Textbook Question

22. Brexit A survey asked 1115 British adults how Britain's decision to leave the European Union has impacted the country. The results are shown in the Pareto chart. Find the

probability of each event. (Adapted from Ipsos)

d. Randomly selecting a British adult who feels that the move has had a fairly positive or very positive impact on Great Britain

Textbook Question

81. Genetics A Punnett square is a diagram that shows all possible gene combinations in a cross of parents whose genes are known. When two pink snapdragon flowers (RW) are crossed, there are four equally likely possible outcomes for the genetic makeup of the offspring: red (RR), pink (RW), pink (WR), and white (WW), as shown in the Punnett square at the left. When two pink snapdragons are crossed, what is the probability that the offspring will be (c) white?

Textbook Question

Politics The responses of 1500 U.S. adults to a survey that asked them to state their own political viewpoints are shown in the Pareto chart. Find the probability of each event.(Adapted from YouGov)

d. Randomly selecting a person from the sample who is either unsure or moderate

Textbook Question

Mental Health A survey asks 4805 parents the severity of the mental issues they experienced from the coronavirus pandemic. The results are shown in the table. A parent is randomly selected from the sample. Find the probability of each event. (Adapted from Kaiser Family Foundation)

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c. The parent did not have major mental health issues or is a mother.

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