Problem

Use the following results from a test for marijuana use, which is provided by a certain drug testing company. Among 148 subjects with positive test results, there are 20 false positive results; among 156 negative results, there are 3 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. (Hint: Construct a table.) The probability that a randomly selected subject tested negative or did not use marijuana is (Do not round until the final answer. Then round to three decimal places as needed.)

Solution

Step 1 :First, we need to construct a table based on the information given in the question. The table will have two rows and two columns. The rows will represent the actual condition of the subjects (whether they used marijuana or not) and the columns will represent the test results (whether the test was positive or negative).

Step 2 :The cell at the intersection of the 'Used Marijuana' row and the 'Test Positive' column will represent the number of true positive results. The cell at the intersection of the 'Used Marijuana' row and the 'Test Negative' column will represent the number of false negative results. The cell at the intersection of the 'Did Not Use Marijuana' row and the 'Test Positive' column will represent the number of false positive results. The cell at the intersection of the 'Did Not Use Marijuana' row and the 'Test Negative' column will represent the number of true negative results.

Step 3 :We know that there are 148 positive test results, 20 of which are false positives. So, there are 148 - 20 = 128 true positives. We also know that there are 156 negative test results, 3 of which are false negatives. So, there are 156 - 3 = 153 true negatives.

Step 4 :The total number of subjects is the sum of true positives, false positives, true negatives, and false negatives, which is 128 + 20 + 153 + 3 = 304.

Step 5 :The probability that a randomly selected subject tested negative or did not use marijuana is the sum of the probabilities that the subject tested negative and that the subject did not use marijuana. The probability that the subject tested negative is the number of negative test results divided by the total number of subjects. The probability that the subject did not use marijuana is the number of subjects who did not use marijuana divided by the total number of subjects. The number of subjects who did not use marijuana is the sum of true negatives and false positives, which is 153 + 20 = 173.

Step 6 :The probability cannot be greater than 1. I made a mistake in my calculations. The probability that a randomly selected subject tested negative or did not use marijuana is not the sum of the probabilities that the subject tested negative and that the subject did not use marijuana. Instead, it is the probability that the subject tested negative plus the probability that the subject did not use marijuana minus the probability that both events occur. The probability that both events occur is the probability that the subject tested negative and did not use marijuana, which is the number of true negatives divided by the total number of subjects.

Step 7 :Final Answer: The probability that a randomly selected subject tested negative or did not use marijuana is \(\boxed{0.579}\).

From Solvely APP
Source: https://solvelyapp.com/problems/17932/

Get free Solvely APP to solve your own problems!

solvely Solvely
Download