Problem 1: (12 points) Screening tests are often used in clinical practice to assess the likelihood that a person has a medical condition. The rationale is that if the disease is identified before symptom onset, then earlier treatment may lead to a cure or improved survival or quality of life. The prostate-specific antigen (PSA) test for prostate cancer measures blood concentrations of PSA, a protein produced by the prostate gland. Elevated levels of PSA may help identify men with prostate cancer. A definitive diagnosis, however, requires biopsies of the prostate gland. A population of men over 50 years old who are considered at high risk for prostate cancer had both the PSA screening test and a biopsy. Among these men, 23% had both elevated PSA levels and a positive biopsy, whereas 50% had low PSA levels and a negative biopsy. 26% had a positive biopsy. 

Draw a venn diagram and write the given information in a set notations. (2 points)



### Step 1: Organize the given information into two sets

Let:
- \( A \) represent the event where a man has elevated PSA levels (positive PSA test).
- \( B \) represent the event where a man has a positive biopsy (indicating prostate cancer).

We are given:
- 23% had both elevated PSA levels and a positive biopsy. This represents the intersection of the sets \( A \) and \( B \), denoted as \( A \cap B \). So, \( P(A \cap B) = 0.23 \).
- 50% had low PSA levels and a negative biopsy. This means the complement of both \( A \) and \( B \), denoted as \( A^c \cap B^c \). So, \( P(A^c \cap B^c) = 0.50 \).
- 26% had a positive biopsy. This represents the probability of \( B \), denoted as \( P(B) = 0.26 \).

### Step 2: Express information in set notation

We can summarize the information as:

- \( P(A \cap B) = 0.23 \)
- \( P(A^c \cap B^c) = 0.50 \)
- \( P(B) = 0.26 \)

### Step 3: Draw a Venn diagram

To represent this in a Venn diagram:
- Draw two intersecting circles, one for \( A \) (elevated PSA levels) and one for \( B \) (positive biopsy).
- Label the intersection of the two circles with \( 23\% \) to indicate \( P(A \cap B) \).
- The total area of the circle representing \( B \) should reflect 26%, which means \( 26\% \) of the population has a positive biopsy.
- The complement area of both circles represents 50%, indicating \( P(A^c \cap B^c) \).

Would you like to go into further detail on solving the problem or explore related probabilities? Here are 5 related questions:

1. How can we determine the probability of having elevated PSA levels, \( P(A) \)?
2. What is the probability that a person has low PSA levels but a positive biopsy?
3. How does this relate to sensitivity and specificity in screening tests?
4. What is the probability of having a negative biopsy regardless of PSA levels?
5. How can we compute the probability of either event \( A \) or \( B \) occurring?

**Tip**: When dealing with probabilities in medical tests, always remember that a positive test result doesn't guarantee the presence of a disease.

Problem 1: (12 points) Screening tests are often used in clinical practice to assess the likelihood that a person has a medical condition. The rationale is that if the disease is identified before symptom onset, then earlier treatment may lead to a cure or improved survival or quality of life. The prostate-specific antigen (PSA) test for prostate cancer measures blood concentrations of PSA, a protein produced by the prostate gland. Elevated levels of PSA may help identify men with prostate cancer. A definitive diagnosis, however, requires biopsies of the prostate gland. A population of men over 50 years old who are considered at high risk for prostate cancer had both the PSA screening test and a biopsy. Among these men, 23% had both elevated PSA levels and a positive biopsy, whereas 50% had low PSA levels and a negative biopsy. 26% had a positive biopsy. Draw a venn diagram and write the given information in a set notations. (2 points)

Learn how to use Venn diagrams and set notation to solve probability problems involving PSA tests and biopsy results. This problem explores key concepts in probability, including complementary events, intersections, and total probabilities. Suitable for high school students in Grades 11-12.

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