Use the following results from a test for marijuana​ use, which is provided by a certain drug testing company. Among 

144144 

subjects with positive test​ results, there are 

2525 

false positive​ results; among 

155155 

negative​ results, there are 

44 

false negative results. If one of the test subjects is randomly​ selected, find the probability that the subject tested negative or did not use marijuana.​ 

To solve this problem, we need to calculate the probability that a randomly selected subject either tested negative or did not use marijuana. Let's break down the information provided and calculate the required probability step by step.

### Step 1: Organize the Data
- **Positive Test Results:** 144 subjects
  - **False Positives:** 25 subjects
- **Negative Test Results:** 155 subjects
  - **False Negatives:** 4 subjects

### Step 2: Calculate True Positives and True Negatives
- **True Positives**: \( 144 - 25 = 119 \)
- **True Negatives**: \( 155 - 4 = 151 \)

### Step 3: Total Number of Subjects
Total number of subjects = \( 144 + 155 = 299 \)

### Step 4: Identify the Events
- **Event A**: The subject tested negative.
- **Event B**: The subject did not use marijuana.

To find the probability that a randomly selected subject tested negative or did not use marijuana, we can use the formula for the union of two events:

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

Where:
- \( P(A) \) is the probability that a subject tested negative.
- \( P(B) \) is the probability that a subject did not use marijuana.
- \( P(A \cap B) \) is the probability that a subject both tested negative and did not use marijuana.

### Step 5: Calculate the Probabilities
1. **Probability that a subject tested negative (P(A)):**
   \[
   P(A) = \frac{\text{Number of negative test results}}{\text{Total number of subjects}} = \frac{155}{299}
   \]

2. **Probability that a subject did not use marijuana (P(B)):**
   - Total number of subjects who did not use marijuana = True Negatives + False Positives = 151 + 25 = 176
   \[
   P(B) = \frac{176}{299}
   \]

3. **Probability that a subject tested negative and did not use marijuana (P(A ∩ B)):**
   - This is simply the probability of a true negative:
   \[
   P(A \cap B) = \frac{151}{299}
   \]

### Step 6: Apply the Formula
Now, substitute the values into the union formula:

\[
P(A \cup B) = \frac{155}{299} + \frac{176}{299} - \frac{151}{299}
\]

\[
P(A \cup B) = \frac{180}{299} \approx 0.602
\]

### Final Answer
The probability that a randomly selected subject either tested negative or did not use marijuana is approximately **0.602** or **60.2%**.

Would you like further details or have any questions?

---

**Related Questions:**
1. What is the probability of a false positive in this test?
2. What is the probability of a true positive in this test?
3. How do false positives and false negatives affect the overall accuracy of a drug test?
4. How can conditional probability be applied to this scenario?
5. How would the probability change if the total number of subjects increased?

**Tip:** When solving probability problems, make sure to clearly identify and calculate the different events and their intersections to avoid confusion.

Calculate the probability that a randomly selected subject tested negative or did not use marijuana based on test results. Understand the concepts of probability, including conditional probability and the union of events formula. Suitable for high school students learning about probability.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation