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In an article that appears on the website of the American Statistical Association, Carlton Gunn, a public defender in Seattle, Washington, wrote about how he uses statistics in his work as an attorney. He states: I personally have used statistics in trying to challenge the reliability of drug testing results. Suppose the chance of a mistake in the taking and processing of a urine sample for a drug test is just 1 in 100 . And your client has a "dirty" (i.e., positive) test result. Only a 1 in 100 chance that it could be wrong? Not necessarily. If the vast majority of all tests given say 99 in 100 -are truly clean, then you get one false dirty and one true dirty in every 100 tests, so that half of the dirty tests are false. Define the following events as \(T D=\) event that the test result is dirty \(T C=\) event that the test result is clean \(D=\) event that the person tested is actually dirty \(C=\) event that the person tested is actually clean a. Using the information in the quote, what are the values of $$ \text { i. } P(T D \mid D) $$ iii. \(P(C)\) $$ \text { ii. } P(T D \mid C) $$ iv. \(P(D)\) b. Use the probabilities from Part (a) to construct a hypothetical 1000 table. c. What is the value of \(P(T D)\) ? d. Use the information in the table to calculate the probability that a person is clean given that the test result is dirty, \(P(C \mid T D)\). Is this value consistent with the argument given in the quote? Explain.

Step by step solution

01

i. Finding P(TD|D)

From the given information, we know that there is a 1 in 100 chance that a drug test comes out dirty by mistake, i.e., given a person is actually clean. Therefore, we know that: $$ P(TD|C) = 0.01 $$ Since it's reasonable to assume that the drug test is reliable when a person is actually dirty, we have: $$ P(TD|D) = 1 $$

02

ii. Finding P(C) and P(D)

The quote provides the information that 99 in 100 tests are clean, i.e., the person tested is actually clean. So, $$ P(C) = \frac{99}{100} = 0.99\\ P(D) = 1-P(C) = 1-0.99 = 0.01 $$ Now, we have: i. \( P(TD|D) = 1 \) ii. \( P(TD|C) = 0.01 \) iii. \( P(C) = 0.99 \) iv. \( P(D) = 0.01 \) #b. Constructing a hypothetical 1000 table#

03

Constructing the hypothetical 1000 table:

Based on the given probabilities, out of 1000 tests, - 990 will be clean (99%) - 10 will be dirty (1%) Out of the 990 clean tests, - 1% will be false dirty tests, i.e., 9.9 false dirty tests - 99% will be true clean tests, i.e., 980.1 true clean tests Out of the 10 dirty tests, - 100% will be true dirty tests, i.e., 10 true dirty tests (Note that 9.9 and 980.1 are numbers using percentages, but we can only have whole numbers of tests, so we approximate by rounding these numbers.) After this correction, we have: - 10 false dirty tests - 980 true clean tests - 10 true dirty tests #c. Finding the value of P(TD)#

04

Finding P(TD)

We can find the probability of a test being dirty (regardless of it being true or false dirty) by dividing the total number of dirty tests by the total number of tests. $$ P(TD) = \frac{10 \text{ false dirty tests} + 10 \text{ true dirty tests}}{1000 \text{ total tests}} = \frac{20}{1000} = 0.02 $$ #d. Calculating P(C|TD)#

05

Finding P(C|TD) using Bayes' Rule

We will use Bayes' rule to find P(C|TD): $$ P(C|TD) = \frac{P(TD|C)P(C)}{P(TD)}\\ P(C|TD) = \frac{0.01 \times 0.99}{0.02} = \frac{0.0099}{0.02} $$ Approximate this value: $$ P(C|TD) \approx 0.5 $$ This value is consistent with the argument given in the quote, as it states that "half of the dirty tests are false," meaning half of the dirty tests are actually clean.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Conditional Probability

Conditional probability is a measure of the likelihood of an event occurring, given that another event has already occurred. In the context of our problem, Carlton Gunn argues that the chance of a drug test being falsely dirty (a false positive) must consider the conditional probability of a test being dirty given the subject is clean.

When we speak about the probability of the test result being dirty given that the actual condition is clean, represented by the symbol \( P(TD|C) \), we're asking, 'If we know the test subject is clean, what are the odds the test says otherwise?' This understanding is vital, especially in law, where the implication of a false positive could significantly affect someone's life. In the given case, the probability that the test shows a dirty result when the person is indeed clean is 1%, or \( P(TD|C) = 0.01 \).

Similarly, understanding the opposite—\( P(TD|D) \)—informs us about the test's accuracy when the subject is actually dirty, which is assumed to be reliable at 100%, or \( P(TD|D) = 1 \). Conditional probabilities allow legal professionals to assess evidence and scenarios critically, and make decisions based on statistical understanding rather than assumptions.

Applying Bayes' Theorem

Bayes' theorem is a way of finding a probability when we know certain other probabilities. The theorem uses the concept of conditional probability to 'reverse' conditional statements. In this case, we want to find the probability that a person is clean given a dirty test result, \( P(C|TD) \).

To find this, we need to know three things:

• The probability that a test reads dirty given the person is clean, \( P(TD|C) \).
• The overall probability of a person being clean, \( P(C) \).
• The probability of obtaining a dirty test result, \( P(TD) \).

Putting these into Bayes' theorem allows us to compute the reversed conditional probability. If half of the tests show a false dirty result as the quote argues, this indicates a need to critically evaluate test reliability and the impact of conditional probabilities on the justice system. The integration of Bayes' theorem in legal cases is an excellent example of applied mathematics in society, showcasing how statistical literacy can provide a different perspective in interpreting evidence.

Assessing Drug Testing Reliability

Drug testing reliability is a crucial concern, especially when such tests are used as evidence in legal contexts. Reliability refers to the consistency and accuracy of the test results. To assess reliability, we must understand both true positive rates (correctly identifying those with drugs) and false positive rates (incorrectly identifying drug-free individuals as drug users).

In this scenario, the reliability of drug tests is challenged through statistical analysis. By constructing a table with 1000 hypothetical tests, we see the expected number of false dirty (positive) and true dirty results. This analysis leads to the startling realization that with a 1% false dirty rate and a low prevalence of actual drug use (1%), about 50% of dirty test results could be false positives. Consequently, evidence based on drug tests alone, without considering the conditional probabilities and false positive rates, could lead to wrongful convictions. Incorporating these statistical tools in legal arguments helps in scrutinizing the evidence and ensuring that justice is served by minimizing the risk of misinterpreting test results.