https://file.aiquickdraw.com/mathbot/file/1727224135820tuuku985.jpg

Let's analyze the problem step-by-step based on the contingency table provided.

### The table:
|                       | Drug User | Not a Drug User | Total  |
|-----------------------|-----------|-----------------|--------|
| **Tests Positive**     | 18        | 2               | 20     |
| **Tests Negative**     | 3         | 45              | 48     |
| **Total**              | 21        | 47              | 68     |

The total population is 68.

### a) **Probability the person is a drug user**

This is a basic probability question. The probability \(P(\text{Drug User})\) is the number of drug users divided by the total number of people.

\[
P(\text{Drug User}) = \frac{\text{Number of Drug Users}}{\text{Total Population}} = \frac{21}{68}
\]

\[
P(\text{Drug User}) = 0.3088 \text{ or } 30.88\%
\]

### b) **Probability the person is a drug user, given that the person tests positive**

This is a conditional probability question. The formula for conditional probability is:

\[
P(\text{Drug User} \mid \text{Tests Positive}) = \frac{P(\text{Drug User and Tests Positive})}{P(\text{Tests Positive})}
\]

From the table:
- The number of people who are drug users and test positive is 18.
- The total number of people who test positive is 20.

So,

\[
P(\text{Drug User} \mid \text{Tests Positive}) = \frac{18}{20} = 0.9 \text{ or } 90\%
\]

### c) **Probability the person tests negative OR is a drug user**

This is a question involving the union of two events. The formula for the union of two events is:

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

Where:
- \(A = \text{Tests Negative}\)
- \(B = \text{Drug User}\)

From the table:
- \(P(\text{Tests Negative}) = \frac{48}{68}\)
- \(P(\text{Drug User}) = \frac{21}{68}\)
- \(P(\text{Tests Negative and Drug User}) = \frac{3}{68}\)

Now we apply the formula:

\[
P(\text{Tests Negative OR Drug User}) = \frac{48}{68} + \frac{21}{68} - \frac{3}{68} = \frac{66}{68} = 0.9706 \text{ or } 97.06\%
\]

### Summary of Answers:
- **a)** \( \frac{21}{68} \) or 30.88%
- **b)** \( \frac{18}{20} \) or 90%
- **c)** \( \frac{66}{68} \) or 97.06%

Would you like further details on any of the steps? 

Here are 5 related questions to deepen your understanding:
1. What is the probability that a person tests positive and is not a drug user?
2. How would the probabilities change if the number of drug users increased?
3. What is the probability that a randomly selected person tests positive, given that they are not a drug user?
4. How can we calculate the false positive rate from the table?
5. What is the complement of the probability in part b?

**Tip:** Conditional probabilities are very useful in medical testing to assess the accuracy of diagnostic tests. Always consider both true and false outcomes.

If one person is randomly selected, find the probability that: a) The person is a drug user. b) The person is a drug user given that the person tests positive. c) The person tests negative OR is a drug user.

Solve a probability problem based on drug test results using a contingency table. Learn how to calculate the probability that a person is a drug user, conditional probability, and the union of events. Suitable for students in Grades 9-12.

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