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?

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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.