### Understanding Drug Test Accuracy with ProbabilitiesA test that screens for illegal drug use, i.e., a “drug test,” is used in a large population of people, 4% of whom actually use drugs. Suppose that the false positive rate in the drug test is 3% and that the false negative rate in the drug test is 2%. Consequently, an individual who actually uses drugs tests positive for drug use 98% of the time, whereas an individual who does not use drugs tests negative for drug use 97% of the time.**Question:**What is the probability that an individual randomly chosen who tests positive for illegal drug use actually uses illegal drugs?This problem involves calculating conditional probability using the given data about the population and the test’s accuracy. Given:- **Prevalence of drug use among the population (P(D)) = 4% = 0.04**- **False positive rate (P(+|ND)) = 3% = 0.03**- **False negative rate (P(-|D)) = 2% = 0.02**From these data points, we can derive:- **True positive rate (P(+|D)) = 98% = 0.98** (since the false negative rate is 2%)- **True negative rate (P(-|ND)) = 97% = 0.97** ### Bayes' Theorem ApplicationTo find the probability that an individual who tests positive actually uses drugs (P(D|+)), we use Bayes’ theorem:\[ P(D|+) = \frac{P(+|D) \cdot P(D)}{P(+)} \]Where:\[ P(+) = P(+|D) \cdot P(D) + P(+|ND) \cdot P(ND) \]\[ P(ND) = 1 - P(D) \]First, calculate P(+):\[ P(+) = (0.98 \times 0.04) + (0.03 \times 0.96) \]\[ P(+) = 0.0392 + 0.0288 \]\[ P(+) = 0.068 \]Now, calculate P(D|+):\[ P(D|+) = \frac{0.98 \times 0.04}{0.068} \]\[ P(D|+) = \

Define the events as follows: D: The selected individual uses drugs Dc: The selected individual does…

A test that screens for illegal drug use, i.e. a“drug test," is used in a large population of people in which 4 % of who actually use drugs. Suppose that the false positive rate in the drug test is 3 % and that the false negative rate in the drug test is 2 %. Accordingly, an individual who actually uses drugs tests positive for drug use 98 % of the time, whereas an individual who does not use drugs tests negative for drug use 97 % of the time. What is the probability that an individual randomly chosen who tests positive for illegal drug use actually uses illegal drugs?

A test that screens for illegal drug use, i.e. a“drug test," is used in a large population of people in which 4 % of who actually use drugs. Suppose that the false positive rate in the drug test is 3 % and that the false negative rate in the drug test is 2 %. Accordingly, an individual who actually uses drugs tests positive for drug use 98 % of the time, whereas an individual who does not use drugs tests negative for drug use 97 % of the time. What is the probability that an individual randomly chosen who tests positive for illegal drug use actually uses illegal drugs?

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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