Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

Arsenic exposure and cancer In a certain county, \(2 \%\) of the people have cancer. Of those with cancer, \(70 \%\) have been exposed to high levels of arsenic. Of those without cancer, \(10 \%\) have been exposed. What percentage of the people who have been exposed to high levels of arsenic have cancer? (Hint: Use a tree diagram.)

Step-by-Step Solution

Answer

14.58% of exposed individuals have cancer.

Step 1: Identify Given Probabilities

Let's restate the given probabilities: - The probability of having cancer, \(P(C)\), is \(0.02\).- The probability of having been exposed to arsenic given that one has cancer, \(P(E|C)\), is \(0.70\).- The probability of having been exposed to arsenic given that one does not have cancer, \(P(E| eg C)\), is \(0.10\).

Step 2: Use Bayes' Theorem

We need to find the probability of having cancer given exposure to arsenic, \(P(C|E)\). This can be calculated using Bayes' Theorem:\[P(C|E) = \frac{P(E|C) \times P(C)}{P(E)}\]

Step 3: Calculate Overall Probability of Exposure

To find \(P(E)\), the probability of exposure to arsenic for anyone, use the law of total probability:\[P(E) = P(E|C) \times P(C) + P(E| eg C) \times P( eg C)\]Substituting the given values:\[P(E) = (0.70 \times 0.02) + (0.10 \times 0.98)\]Calculate this to find \(P(E)\).

Step 4: Compute Specific Probability Using Bayes’ Theorem

Substitute the result from Step 3 into Bayes' Theorem:\[P(C|E) = \frac{0.70 \times 0.02}{P(E)}\]Calculate \(P(C|E)\) to find the percentage of people exposed to arsenic who also have cancer.

Step 5: Convert to Percentage

Multiply the probability \(P(C|E)\) obtained in Step 4 by 100 to express it as a percentage. This gives us the percentage of exposed individuals who have cancer.

Key Concepts

Probability

Probability is a numerical measure of the likelihood of a particular event occurring. It ranges from 0 to 1, where 0 indicates that the event is impossible, and 1 suggests certainty.
In this exercise, several probabilities are given:

• The probability that a person has cancer, denoted as \( P(C) \), is 0.02, meaning 2% of people in the county have cancer.
• Conversely, the probability that a person does not have cancer, denoted as \( P(eg C) \), can be calculated as \( 1 - P(C) = 0.98 \).
• Additionally, there are conditional probabilities that describe the likelihood of being exposed to arsenic depending on the presence of cancer.

Understanding these basic probabilities is critical for analyzing more complex relationships in the data.

Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. For example, in this exercise, we examine

• The probability of being exposed to arsenic given that a person has cancer, denoted as \( P(E|C) = 0.70 \). This indicates that 70% of those with cancer have been exposed to high levels of arsenic.
• The probability of arsenic exposure given no cancer is present, \( P(E| eg C) = 0.10 \), meaning 10% of those without cancer have been exposed.

Bayes' Theorem famously uses these types of probabilities for calculating inverse probabilities, allowing us to find \( P(C|E) \), the probability of having cancer given exposure.

Tree Diagram

A tree diagram is a useful visual tool for laying out all possible outcomes of an event and their associated probabilities. This exercise involves two main events: having cancer or not, and arsenic exposure or not.
Each branch of the tree diagram represents a possible outcome, and by following the branches, we can calculate composite probabilities. Here’s how we can use it:

• Start with the initial probabilities (e.g., having cancer and not having cancer).
• Extend branches to arsenic exposure and no exposure for each starting probability.
• Multiply down the branches to find combined probabilities (e.g., \( P(E \cap C) \) for exposure and cancer).

Tree diagrams not only organize information clearly, but they also help in applying the law of total probability for complete probability analysis.

Law of Total Probability

The law of total probability helps calculate the total likelihood of an outcome that can be realized in different ways. It’s essential in this problem for determining the overall probability of arsenic exposure, \( P(E) \). According to the law, we consider every way arsenic exposure could occur: workers with cancer and those without.
Here's the breakdown:

• The term \( P(E) \) is split into two scenarios: one where individuals have cancer and are exposed (\( P(E|C) \times P(C) \)) and one where they do not have cancer but still are exposed (\( P(E| eg C) \times P(eg C) \)).
• Mathematically, this means combining both probabilities: \[ P(E) = P(E|C) \times P(C) + P(E| eg C) \times P(eg C) \].

This overarching probability value is then utilized in Bayes' Theorem to find \( P(C|E) \). Using this law allows us to piece together different probabilities into a complete picture of potential exposure scenarios.