Ch. 3 - Probability

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 3 - Probability

Problem 3.2.21a

Chapter 3, Problem 3.2.21a

"Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
21. BRCA1 Gene Research has shown that approximately 1 woman in 400 carries a mutation of the BRCA1 gene. About 64% of women with this mutation develop breast
cancer. Find the probability that a randomly selected woman will carry the mutation of the BRCA1 gene and will develop breast cancer. (Source: National Cancer Institute)

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the probability that a randomly selected woman both carries the BRCA1 gene mutation and develops breast cancer. This is a joint probability, which can be calculated using the Multiplication Rule.

Step 2: Recall the Multiplication Rule for probabilities. The rule states that the probability of two events A and B occurring together (P(A and B)) is equal to the probability of A occurring (P(A)) multiplied by the probability of B occurring given that A has occurred (P(B|A)).

Step 3: Identify the given probabilities. From the problem, we know: P(A) = Probability of carrying the BRCA1 gene mutation = 1/400, and P(B|A) = Probability of developing breast cancer given the mutation = 64% or 0.64.

Step 4: Apply the Multiplication Rule. Substitute the values into the formula: P(A and B) = P(A) × P(B|A). This will give the probability of a woman carrying the mutation and developing breast cancer.

Step 5: Interpret the result. Once calculated, the result will represent the likelihood of a randomly selected woman both carrying the BRCA1 gene mutation and developing breast cancer. Ensure the units and context are clear.

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Key Concepts

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

Multiplication Rule

The Multiplication Rule in probability states that the probability of two independent events occurring together is the product of their individual probabilities. In this context, it helps calculate the likelihood of a woman both carrying the BRCA1 mutation and developing breast cancer by multiplying the probability of each event.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has already occurred. In this scenario, the probability of developing breast cancer is conditional on having the BRCA1 mutation, which is crucial for accurately determining the overall probability of both events happening.

Sample Space

The sample space is the set of all possible outcomes of a probability experiment. In this case, the Venn diagram illustrates the sample space of women, highlighting those with the BRCA1 mutation and those who develop breast cancer, which aids in visualizing the relationships between these two groups and calculating the desired probabilities.

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Related Practice

Textbook Question

"Finding Conditional Probabilities In Exercises 7 and 8, use the table to find each conditional probability.

8. Retirement Savings The table shows the results of a survey in which 250 male and 250 female workers ages 25 to 64 were asked if they contribute to a retirement savings plan at

work.

a. Find the probability that a randomly selected worker contributes to a retirement savings plan at work, given that the worker is male.

Textbook Question

"39. Reliability of Testing A virus infects one in every 200 people. A test used to detect the virus in a person is positive 80% of the time when the person has the virus and 5% of the time when the person does not have the virus. (This 5% result is called a false positive.) Let A be the event ""the person is infected"" and B be the event ""the person tests positive.""

a. Using Bayes' Theorem, when a person tests positive, determine the probability that the person is infected."

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Textbook Question

50. Investment Committee A company has 200 employees, consisting of 144 women and 56 men. The company wants to select five employees to serve as an investment committee.

a. Use technology to find the number of ways that 5 employees can be selected from 200.

Textbook Question

Finding Conditional Probabilities In Exercises 7 and 8, use the table to find each conditional probability.

7. Business Degrees The table shows the numbers of male and female students in the United States who received bachelor's degrees in business and nonbusiness fields in a recent year. (Source: National Center for Educational Statistics)

a. Find the probability that a randomly selected bachelor's degree-earning student is male, given that the degree is in business.

Textbook Question

87. College Football A stem-and-leaf plot for the numbers of touchdowns allowed by the 127 NCAA Division I Football Bowl Subdivision teams in the 2020-2021 season is shown. Find the probability that a team chosen at random allowed (a) at least 51 touchdowns. Are any of these events unusual? Explain. (Source: National Collegiate Athletic Association)

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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)

a. Find the probability that all three adult U.S. citizens say that Donald Trump was the worst president in U.S. history."