The following contingency table summarizes the drug test results for 383 test-takers:
 	Positive	Negative	Total
Doesn't use drugs	2	161	
163
Correct
Uses drugs	212	8	
220
Correct
Total	
214
Correct	
169
Correct	
383
Correct
For a randomly selected test administration, find the following probabilities. (Round the answers to 4 decimal places.)

1) The probability that the test result is positive:

 

2) The probability of a false positive:

 

3) The probability of an error:



Let's analyze the given contingency table:

|                          | Positive | Negative | Total |
|--------------------------|----------|----------|-------|
| **Doesn't use drugs**     | 2        | 161      | 163   |
| **Uses drugs**            | 212      | 8        | 220   |
| **Total**                 | 214      | 169      | 383   |

From this table, we can calculate the required probabilities.

### 1) The probability that the test result is positive:
The probability that the test result is positive is given by the ratio of the total number of positive results to the total number of test-takers:

\[
P(\text{Positive}) = \frac{\text{Total Positive Results}}{\text{Total Test-takers}} = \frac{214}{383}
\]

Let's calculate:

\[
P(\text{Positive}) = \frac{214}{383} \approx 0.5585
\]

### 2) The probability of a false positive:
A false positive occurs when the test result is positive, but the person does not use drugs. This is given by the ratio of the number of false positives (people who don't use drugs but test positive) to the total number of test-takers:

\[
P(\text{False Positive}) = \frac{\text{False Positives (2)}}{\text{Total Test-takers}} = \frac{2}{383}
\]

Let's calculate:

\[
P(\text{False Positive}) = \frac{2}{383} \approx 0.0052
\]

### 3) The probability of an error:
An error occurs either from a false positive or a false negative. Therefore, the probability of an error is the sum of the probabilities of a false positive and a false negative. A false negative occurs when the test result is negative, but the person uses drugs. So we sum the two errors:

\[
P(\text{Error}) = \frac{\text{False Positives (2)} + \text{False Negatives (8)}}{\text{Total Test-takers}} = \frac{2 + 8}{383}
\]

Let's calculate:

\[
P(\text{Error}) = \frac{10}{383} \approx 0.0261
\]

### Summary of probabilities:
1. **Probability that the test result is positive:** 0.5585
2. **Probability of a false positive:** 0.0052
3. **Probability of an error:** 0.0261

Would you like further details or clarification on any part?

Here are 5 related questions for further practice:
1. What is the probability that the test result is negative?
2. What is the probability that the test result is negative given that the person uses drugs?
3. What is the probability of a false negative?
4. What is the probability that the person uses drugs given a positive result?
5. What is the probability that the person does not use drugs given a negative result?

**Tip:** Probabilities related to errors are often essential in medical testing to assess the accuracy of diagnostic tools.

The following contingency table summarizes the drug test results for 383 test-takers:

Positive	Negative	Total
Doesn't use drugs	2	161	163
Uses drugs	212	8	220
Total	214	169	383

For a randomly selected test administration, find the following probabilities. (Round the answers to 4 decimal places.)

1) The probability that the test result is positive:

2) The probability of a false positive:

3) The probability of an error:

Learn how to calculate key probabilities such as the probability of a positive result, a false positive, and an error using a drug test contingency table for 383 test-takers. This problem involves probability concepts and the use of contingency tables, suitable for students in Grades 9-12.

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