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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

The distribution of National Honor Society members among the students at a local high school is shown in the table. A student's name is drawn at random. $$ \begin{array}{lcc} \text { Class } & \text { Total } & \text { National Honor Society } \\ \hline \text { Senior } & 92 & 37 \\ \hline \text { Junior } & 112 & 30 \\ \hline \text { Sophomore } & 125 & 20 \\ \hline \text { Freshman } & 120 & 0 \\ \hline \end{array} $$ (a) What is the probability that the student is a junior? (b) What is the probability that the student is a senior, given that the student is in the National Honor Society?

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How many different simple random samples of size 5 can be obtained from a population whose size is 50?

According to a Gallup Poll, about \(17 \%\) of adult Americans bet on professional sports. Census data indicate that \(48.4 \%\) of the adult population in the United States is male. (a) Assuming that betting is independent of gender, compute the probability that an American adult selected at random is male and bets on professional sports. (b) Using the result in part (a), compute the probability that an American adult selected at random is male or bets on professional sports. (c) The Gallup poll data indicated that \(10.6 \%\) of adults in the United States are males and bet on professional sports. What does this indicate about the assumption in part (a)? (d) How will the information in part (c) affect the probability you computed in part (b)?
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