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

The table:

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.