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Chapter 5: Problem 14

According to the U.S. National Center for Health Statistics, in \(2002,0.2 \%\) of deaths in the United States were 25 - to 34 -year-olds whose cause of death was cancer. In addition, \(1.97 \%\) of all those who died were 25 to 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 to 34 years old?

Short Answer

Expert verified
The probability is approximately 10.15%.

Step by step solution

01

- Understand the given data

Given that 0.2% of deaths were 25-34 year olds who died from cancer. Additionally, 1.97% of all deaths were 25-34 year olds.
02

- Define Events

Let A be the event that the death was due to cancer, and B be the event that the individual was 25-34 years old.We need to find the probability of A given B, denoted as P(A|B).
03

- Use Conditional Probability Formula

The formula for conditional probability is: \[ P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \] Here, P(A and B) is the probability that the death was due to cancer and the individual was 25-34 years old, and P(B) is the probability that the individual was 25-34 years old.
04

- Set the values

From the given data, P(A and B) = 0.2% = 0.002 and P(B) = 1.97% = 0.0197.
05

- Calculate the Conditional Probability

Using the given values in the conditional probability formula:\[ P(A|B) = \frac{0.002}{0.0197} \] Calculate this to get P(A|B).
06

- Simplify the fraction

\[ P(A|B) = \frac{0.002}{0.0197} \approx 0.1015 \] Thus, the probability that the death was due to cancer given that the individual was 25-34 years old is approximately 0.1015 or 10.15%.

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Key Concepts

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

Probability Theory
Probability theory is a branch of mathematics focused on the analysis of random events. It deals with the likelihood that a given event will occur. In this context, we're interested in conditional probability, which evaluates the probability of an event, given that another event has already occurred. This is crucial in many real-world scenarios where we have to adjust our calculations based on new information. For example, finding the likelihood of a medical condition in a specific age group often involves conditional probabilities.
Event Definition
To solve any probability problem, it's essential to define the events clearly. In our exercise, we have two events: Event A and Event B. Event A is the event where the individual died from cancer, and Event B is the event where the individual was between 25 to 34 years old. Defining these events helps clarify what we are calculating: the probability of cancer being the cause of death, given the individual fell within a certain age range. This step lays the groundwork for applying the right mathematical formulas later.
Statistical Analysis
Statistical analysis involves collecting and interpreting data to identify patterns and trends. In our exercise, we use provided statistical data about deaths in the U.S. in 2002 to carry out the analysis. The key values given are the overall percentage of deaths in the 25-34 age group (1.97%) and the percentage of those deaths that were due to cancer (0.2%). By leveraging these statistics and utilizing the conditional probability formula, we can determine the probability we鈥檙e interested in. It鈥檚 about choosing the right data and correctly applying statistical methods.
Percentage Calculation
Percentage calculations are vital in making sense of statistical data. In this problem, we convert percentages to their decimal forms to make the calculations easier. We see that 0.2% translates to 0.002 and 1.97% translates to 0.0197. Calculating the conditional probability becomes a matter of dividing these values, which requires basic percentage knowledge but provides meaningful insights. The final step simplifies our fraction to provide a much more comprehensible probability figure. In this case, approximately 10.15%, making the interpretation of the result straightforward for decision-making or further analysis.

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