Prostate cancer is the most common type of cancer found in males. As an indicator of whether a male has prostate cancer, doctors often perform a test that measures the level of the prostate-specific antigen (PSA) that is produced only by the prostate gland. Although PSA levels are indicative of cancer, the test is notoriously unreliable.

The probability that a noncancerous man will have an elevated PSA level is approximately 0.135, increasing to approximately 0.268 if the man does have cancer.

If, based on other factors, a doctor is 90 % certain that a male has prostate cancer, what is the probability that he has the cancer, given that

(Round your answers to three decimal places.)

 

a. the test indicated an elevated PSA level?

To solve this problem, we can use Bayes' theorem, which is helpful in finding conditional probabilities. In this context, Bayes' theorem will help us determine the probability that a man has prostate cancer given that his PSA test result is elevated.

### **Bayes' Theorem**

Bayes' theorem states:

\[
P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}
\]

Where:

- \( P(A|B) \): Probability of event \( A \) given event \( B \).
- \( P(B|A) \): Probability of event \( B \) given event \( A \).
- \( P(A) \): Probability of event \( A \).
- \( P(B) \): Probability of event \( B \).

### **Applying to the Problem**

Let:

- \( A \) = Man has prostate cancer.
- \( B \) = Elevated PSA level.

From the problem, we know:

- Probability the man has cancer: \( P(A) = 0.90 \).
- Probability of elevated PSA if the man has cancer: \( P(B|A) = 0.268 \).
- Probability of elevated PSA if the man does not have cancer: \( P(B|A') = 0.135 \).
- Probability the man does not have cancer: \( P(A') = 1 - 0.90 = 0.10 \).

### **Finding \( P(B) \)**

\( P(B) \) is the total probability of having an elevated PSA level, which can be found using the law of total probability:

\[
P(B) = P(B|A) \cdot P(A) + P(B|A') \cdot P(A')
\]

Substituting the values:

\[
P(B) = (0.268 \times 0.90) + (0.135 \times 0.10)
\]

Let's calculate this:

\[
P(B) = (0.268 \times 0.90) + (0.135 \times 0.10) = 0.2412 + 0.0135 = 0.2547
\]

### **Finding \( P(A|B) \)**

Now, using Bayes' theorem:

\[
P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{0.268 \times 0.90}{0.2547}
\]

Let's calculate this:

\[
P(A|B) = \frac{0.268 \times 0.90}{0.2547} = \frac{0.2412}{0.2547} \approx 0.947
\]

### **Answer**

**The probability that the man has prostate cancer given that his PSA level is elevated is approximately 0.947.**

Would you like more details or have any questions?

---

Here are some related questions:

1. What is the interpretation of \( P(A|B) \) in medical testing?
2. How would the probability change if the prior probability of cancer was different?
3. What happens to the probability if the test reliability improves?
4. How can the probability help in clinical decision-making?
5. How does the prevalence of the disease in the population affect these probabilities?

**Tip:** Bayes' theorem is a powerful tool in diagnosing diseases where test accuracy and initial beliefs about the condition are known.

Prostate cancer is the most common type of cancer found in males. As an indicator of whether a male has prostate cancer, doctors often perform a test that measures the level of the prostate-specific antigen (PSA) that is produced only by the prostate gland. Although PSA levels are indicative of cancer, the test is notoriously unreliable. The probability that a noncancerous man will have an elevated PSA level is approximately 0.135, increasing to approximately 0.268 if the man does have cancer. If, based on other factors, a doctor is 90% certain that a male has prostate cancer, what is the probability that he has the cancer, given that the test indicated an elevated PSA level?

Solve a probability problem using Bayes' theorem to determine the likelihood of prostate cancer given an elevated PSA level. This example uses the law of total probability and applies conditional probabilities to a real-world medical context. Suitable for advanced high school or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation