Question

use the following results from a test for marijuana use, which is provided by a certain drug testing company among 145 subjects with positive test results, there are 20 false positive results; among 151 negative results, there are 5 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.)

Explanation:

Step1: Construct a contingency table

First, we define the categories:

• True positive (TP): Subjects who used marijuana and tested positive. Among 145 positive results, 20 are false positive, so \( TP = 145 - 20 = 125 \)
• False positive (FP): 20 (used no marijuana, tested positive)
• True negative (TN): Subjects who used no marijuana and tested negative. Among 151 negative results, 5 are false negative, so \( TN = 151 - 5 = 146 \)
• False negative (FN): 5 (used marijuana, tested negative)

Now, the contingency table is:

	Used Marijuana	Did Not Use Marijuana	Total
Test Negative	5	146	151
Total	\( 125 + 5 = 130 \)	\( 20 + 146 = 166 \)	\( 145 + 151 = 296 \)

Step2: Use the formula for probability of union

The formula for \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), where:

• \( A \): Tested negative
• \( B \): Did not use marijuana

First, find \( P(A) \): Number of subjects who tested negative is 151, total subjects is 296, so \( P(A) = \frac{151}{296} \)

Second, find \( P(B) \): Number of subjects who did not use marijuana is 166, so \( P(B) = \frac{166}{296} \)

Third, find \( P(A \cap B) \): Number of subjects who tested negative and did not use marijuana is 146, so \( P(A \cap B) = \frac{146}{296} \)

Now, calculate \( P(A \cup B) \):

\( P(A \cup B) = \frac{151}{296} + \frac{166}{296} - \frac{146}{296} \)

\( = \frac{151 + 166 - 146}{296} \)

\( = \frac{171}{296} \)

Step3: Calculate the decimal value

\( \frac{171}{296} \approx 0.5777 \) (rounded to four decimal places for intermediate step), then round to three decimal places: \( 0.578 \)

Answer:

\( 0.578 \)