https://file.aiquickdraw.com/mathbot/file/1729817175281mzlf952n.jpg

The problem is asking for the probability that a randomly chosen person has cancer given that the test indicates cancer, which is a conditional probability problem. Let's break this down:

We are asked to find:

\[
P(\text{Cancer} \mid \text{Test positive})
\]

Using **Bayes' Theorem**, we can calculate this as:

\[
P(\text{Cancer} \mid \text{Test positive}) = \frac{P(\text{Test positive} \mid \text{Cancer}) \cdot P(\text{Cancer})}{P(\text{Test positive})}
\]

### Information given:
1. \( P(\text{Cancer}) = 0.01 \) (1% of the sample has cancer).
2. \( P(\text{No cancer}) = 0.99 \) (the remaining 99% do not have cancer).
3. \( P(\text{Test positive} \mid \text{Cancer}) = 0.99 \) (the test indicates cancer in 99% of those who have cancer).
4. \( P(\text{Test positive} \mid \text{No cancer}) = 0.01 \) (the test indicates cancer in 1% of those who do not have cancer).

### Step 1: Calculate \( P(\text{Test positive}) \)
This can be found by considering both those with cancer and those without cancer who test positive:

\[
P(\text{Test positive}) = P(\text{Test positive} \mid \text{Cancer}) \cdot P(\text{Cancer}) + P(\text{Test positive} \mid \text{No cancer}) \cdot P(\text{No cancer})
\]

Substitute the values:

\[
P(\text{Test positive}) = (0.99 \cdot 0.01) + (0.01 \cdot 0.99)
\]
\[
P(\text{Test positive}) = 0.0099 + 0.0099 = 0.0198
\]

### Step 2: Apply Bayes' Theorem
Now, we can calculate the conditional probability:

\[
P(\text{Cancer} \mid \text{Test positive}) = \frac{0.99 \cdot 0.01}{0.0198} = \frac{0.0099}{0.0198} \approx 0.500
\]

Thus, the probability that a person has cancer given that the test indicates cancer is approximately **0.500** (rounded to three decimal places).

Would you like further clarification on any part? Here are five related questions to deepen understanding:

1. How would the result change if the test's accuracy for non-cancer patients improved?
2. How can Bayes' Theorem be applied in medical diagnostics for other conditions?
3. What is the probability of a person not having cancer given a negative test result?
4. How does changing the base rate (percentage of people with cancer) affect the result?
5. Can you calculate \( P(\text{No cancer} \mid \text{Test positive}) \)?

**Tip:** In probability problems, always identify if you need to use Bayes' Theorem to solve for conditional probabilities when multiple conditions are involved.

Based on these results, what is the probability of a randomly chosen person having cancer given that the test indicates cancer?

Learn how to use Bayes' Theorem to calculate the probability of having cancer given a positive test result. This problem involves understanding conditional probability and calculating P(Cancer | Test positive) using key information. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation