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Chapter 5: Problem 14

According to the U.S. National Center for Health Statistics, \(0.15 \%\) of deaths in the United States are 25 - to 34-year-olds whose cause of death is cancer. In addition, \(1.71 \%\) of all those who die are \(25-34\) years old. What is the probability that a randomly selected death is the result of cancer if the individual is known to have been \(25-34\) years old?

Short Answer

Expert verified
The probability is approximately \(0.0877\).

Step by step solution

01

- Understand the given probabilities

According to the problem, \(0.15 \%\) of deaths are 25-34-year-olds whose cause of death is cancer. This means the probability that a death is due to cancer and the person was 25-34 years old is \(P(C \cap A) = 0.0015\). In addition, \(1.71 \%\) of all those who die are 25-34 years old, so the probability that a randomly selected death is a 25-34-year-old is \(P(A) = 0.0171\).
02

- Use Bayes' Theorem

We want to find the probability that a death is due to cancer given that the individual was 25-34 years old. This can be calculated using Bayes' Theorem: \[P(C|A) = \frac{P(C \cap A)}{P(A)}\]
03

- Plug in the given probabilities

Substitute the given probabilities into the formula from Step 2:\[P(C|A) = \frac{P(C \cap A)}{P(A)} = \frac{0.0015}{0.0171}\]
04

- Simplify the fraction

Simplify the fraction to find the probability:\[P(C|A) = \frac{0.0015}{0.0171} \approx 0.0877\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bayes' Theorem
Bayes' Theorem is a fundamental principle in probability theory and statistical inference. It describes the probability of an event, based on prior knowledge of conditions that might be related to the event. The theorem can be expressed through the formula: \ \(P(A|B) = \frac{P(B|A) \, P(A)}{P(B)}\). In our exercise, we want to know what's the likelihood of a cancer-related death given the individual was 25-34 years old. The formula helps us reverse the conditioning by using known probabilities. This theorem is crucial for updating the probability estimates based on new data, making it widely used in various fields like diagnostics, finance, and machine learning.
Conditional Probability
Conditional probability is the likelihood of an event occurring given that another event has already happened. This is represented as \ \(P(A|B)\). In our context, it means we are trying to find the probability that the cause of death was cancer (Event C) given that the individual was 25-34 years old (Event A). Knowing how to calculate conditional probability is essential in many real-world problems, such as medical diagnoses or quality control in manufacturing.
Probability Rules
Understanding basic probability rules is essential for handling more complicated problems. One key rule for our exercise is the joint probability, represented as \ \(P(A \cap B)\), which is the probability that both events A and B occur. Also crucial is the law of total probability, which provides a way to break down complex probabilities into simpler parts. In our exercise, we use these rules to break down and simplify the problem into manageable pieces before applying Bayes' Theorem.
Statistical Analysis
Statistical analysis involves collecting, organizing, and interpreting data to make informed decisions. In the context of our problem, we use statistical methods to sift through the data provided, such as the percentages of different causes of death in specific age groups. This type of analysis allows us to calculate the probability of a cancer-related death in young adults using Bayes' Theorem. Mastering statistical analysis provides valuable insights and allows for data-driven decision making, important in many fields ranging from healthcare to business.
Real-World Applications
Bayes' Theorem and concepts like conditional probability have many practical applications. For example, in healthcare, they are used to update the probability of a disease based on test results. In finance, they help calculate risks and returns on investments. In our exercise, these concepts allow us to determine the likelihood of a cancer-related death in 25-34-year-olds, which can be useful for public health planning and resource allocation. The power of these statistical tools lies in their ability to incorporate new data and refine probability assessments, making them indispensable in various real-world scenarios.

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Most popular questions from this chapter

For a parallel structure of identical components, the system can succeed if at least one of the components succeeds. Assume that components fail independently of each other and that each component has a 0.15 probability of failure. (a) Would it be unusual to observe one component fail? Two components? (b) What is the probability that a parallel structure with 2 identical components will succeed? (c) How many components would be needed in the structure so that the probability the system will succeed is greater than \(0.9999 ?\)

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Players in sports are said to have "hot streaks" and "cold streaks." For example, a batter in baseball might be considered to be in a slump, or cold streak, if he has made 10 outs in 10 consecutive at-bats. Suppose that a hitter successfully reaches base \(30 \%\) of the time he comes to the plate. (a) Find and interpret the probability that the hitter makes 10 outs in 10 consecutive at-bats, assuming that at-bats are independent events. Hint: The hitter makes an out \(70 \%\) of the time. (b) Are cold streaks unusual? (c) Find the probability the hitter makes five consecutive outs and then reaches base safely. (d) Discuss the assumption of independence in consecutive at-bats.

According to the Centers for Disease Control, the probability that a randomly selected citizen of the United States has hearing problems is \(0.151 .\) The probability that a randomly selected citizen of the United States has vision problems is \(0.093 .\) Can we compute the probability of randomly selecting a citizen of the United States who has hearing problems or vision problems by adding these probabilities? Why or why not?

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