91Ó°ÊÓ

In an article that appears on the web site of the American Statistical Association (www.amstat.org), 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, and \(C=\) event that the person tested is actually clean. a. Using the information in the quote, what are the values of \mathbf{i} . ~ \(P(T D \mid D)\) iii. \(P(C)\) ii. \(P(T D \mid C) \quad\) iv. \(P(D)\) b. Use the law of total probability to find \(P(T D)\). c. Use Bayes' rule to evaluate \(P(C \mid T D)\). Is this value consistent with the argument given in the quote? Explain.

Step by step solution

01

Assign Values

From the statement, we know that:\n- \(P(T D | D) = 1\) (if person is dirty, the test is always dirty), \n- \(P(T D | C) = 0.01\) (there is a 1% chance that a clean person will test dirty), \n- \(P(C) = 0.99\) (99% of people tested are clean), \n- \(P(D) = 0.01\) (1% of people tested are dirty).

02

Calculate Total Probability

We can use the law of total probability to calculate \(P(T D)\). The total probability law states that the probability of event A is equal to the sum of the probability of A intersecting with each event Bi in a partition of the sample space.\nSo, \(P(T D) = P(T D | D) * P(D) + P(T D | C) * P(C) = 1 * 0.01 + 0.01 * 0.99 = 0.02.\)

03

Use Bayes' Rule

Bayes' Rule is used to reverse conditional probabilities. The formula is: \n\(P(A | B) = P(B | A) * P(A) / P(B)\)\nWe need to find \(P(C | T D)\), the probability that a person is clean given a dirty test.\nSo, \(P(C | T D) = P(T D | C) * P(C) / P(T D) = 0.01 * 0.99 / 0.02 = 0.495\)\nThis indicates that there is a 49.5% chance that a person is clean even if the test is dirty. This result is consistent with the public defender's statement that 'half of the dirty tests are false'.

04

Interpretation

The public defender's statistical argument checks out mathematically. For a test that only makes a mistake 1% of the time, if 99% of the people are clean, then about 50% of the positive tests could be false positives. It is important to consider the base rates, in this case the fact that a very large majority of the tested people are clean.

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Key Concepts

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

Law of Total Probability

The Law of Total Probability is a crucial concept in probability theory. It helps us determine the probability of a certain event when there are multiple ways for it to occur. Imagine you're trying to find out how likely it is for a test to show a dirty result, considering both scenarios: whether the person is actually dirty (has used drugs) or clean.This law tells us that the probability of any event (say "a dirty test result") is the sum of the probabilities of that event intersecting with each possible outcome. Here's how it works in our example:

• Probability that a person is dirty and gets a dirty test: this is 1%.
• Probability that a person is clean but still gets a dirty test result: this is 0.99% of the time.

We calculate the total probability by adding these two paths: \[P(T D) = P(T D \mid D) \cdot P(D) + P(T D \mid C) \cdot P(C)\]Using the specific probabilities given in the exercise:\[P(T D) = (1 \cdot 0.01) + (0.01 \cdot 0.99) = 0.02\]This calculation shows that there is a 2% chance of getting a dirty test result overall.

Bayes' Rule

Bayes' Rule is a powerful tool for updating the probability of a hypothesis as more evidence or information becomes available. It's used to "flip" conditional probabilities. That means it's highly useful when you need to determine the likelihood of an initial assumption or cause given an outcome.Let's say you have a dirty test result and you want to know, given this dirtiness, how likely the person is actually clean. The formula for Bayes' Rule is:\[P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}\]In our example, it helps to find:

• The probability of a clean person, given they've received a dirty test result (\(P(C \mid T D)\)).

Plugging in the values from the exercise:\[P(C \mid T D) = \frac{P(T D \mid C) \cdot P(C)}{P(T D)} = \frac{0.01 \cdot 0.99}{0.02} = 0.495\]This means there is approximately a 49.5% chance that a person is clean even though the test result shows dirty, perfectly aligning with the public defender’s argument.

False Positive Rate

Understanding the False Positive Rate is vital, particularly in testing scenarios. A false positive occurs when a test incorrectly indicates the presence of a condition (such as drug use) when it isn't there. In our given example, it's when a clean person returns a dirty test. The false positive rate can mislead interpretations of test accuracy, especially if a test is widely used among a majority group that doesn’t possess the condition, just like our exercise where 99% of people tested are actually clean. Why is this crucial?

• It teaches us that even tests with a low error rate can produce a lot of false positives if the base rate (actual occurrence of condition within the population) is significantly low.
• In this scenario, the false positive rate is a key factor that resulted in about half the positive test results being wrong.

Highlighting the importance of reviewing not just the error rate of tests, but the broader context of testing populations to truly understand the test's effectiveness.