Problem 16 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

You are diagnosed with an uncommon disease. You know that there only is a \(1 \%\) chance of getting it. Use the letter \(D\) for the event "you have the disease" and \(T\) for "the test says so." It is known that the test is imperfect: \(\mathrm{P}(T \mid D)=0.98\) and \(\mathrm{P}\left(T^{c} \mid D^{c}\right)=0.95\). a. Given that you test positive, what is the probability that you really have the disease? b. You obtain a second opinion: an independent repetition of the test. You test positive again. Given this, what is the probability that you really have the disease?

Step-by-Step Solution

Answer

a) ~16.5%, b) ~79.4%.

Step 1: Understanding the problem

The problem provides probabilities related to a rare disease and the accuracy of a medical test. We need to find the probability of actually having the disease given the test results.

Step 2: Known probabilities and terms

Let's define and list all known probabilities:- Probability of having the disease, \( P(D) = 0.01 \)- Probability of not having the disease, \( P(D^c) = 0.99 \)- Probability of testing positive given having the disease, \( P(T | D) = 0.98 \)- Probability of testing negative given not having the disease, \( P(T^c | D^c) = 0.95 \)- Probability of testing positive given not having the disease, \( P(T | D^c) = 1 - P(T^c | D^c) = 0.05 \)

Step 3: Apply Bayes' Theorem for first test

Use Bayes' theorem to find \( P(D | T) \):\[ P(D | T) = \frac{P(T | D) P(D)}{P(T)} \]We need \( P(T) \), calculated as:\[ P(T) = P(T | D) P(D) + P(T | D^c) P(D^c) = 0.98 \cdot 0.01 + 0.05 \cdot 0.99 \]Thus, \( P(T) = 0.0098 + 0.0495 = 0.0593 \).Applying to Bayes' theorem:\[ P(D | T) = \frac{0.98 \times 0.01}{0.0593} \approx 0.165 \]

Step 4: Probability after the second test

For the second opinion, update the probability using the result of the second independent test:Set initial probabilities: \( P(D) = 0.165 \) and \( P(D^c) = 1 - P(D) = 0.835 \).Re-use Bayes' Theorem with updated \( P(T) \):\[ P(D | T, T') = \frac{P(T | D) P(D)}{P(T)} \]Recalculate \( P(T) \):\[ P(T) = 0.98 \times 0.165 + 0.05 \times 0.835 = 0.1617 + 0.04175 = 0.20345 \]\[ P(D | T, T') = \frac{0.98 \times 0.165}{0.20345} \approx 0.794 \]

Step 5: Conclusion

After one positive test, the probability of having the disease is about 16.5%. After a second independent positive test, this probability increases to approximately 79.4%.

Key Concepts

Understanding Conditional Probability

Conditional probability is a key concept in probability theory and is especially useful in real-world scenarios like disease testing. It helps us know the probability of an event happening, given that another event has already occurred. In this context, it gauges how likely it is that a person has a disease given the results of two consecutive tests.

The formula to calculate conditional probability is straightforward:

• If you want to know the probability of event A happening given that event B has happened (denoted as \( P(A | B) \)), you can use Bayes' Theorem.

Bayes' theorem for conditional probability is stated as follows:\[ P(A | B) = \frac{P(B | A) P(A)}{P(B)} \] Here,

• \( P(B | A) \) is the probability of event B occurring given that event A is true.
• \( P(A) \) is the probability of event A occurring by itself.
• \( P(B) \) is the total probability of event B occurring.

In the disease testing problem above, by substituting values for the initial test and similarly for the second test, we calculated how the likelihood changes with additional evidence.

Disease Testing and Its Complexities

Disease testing, while crucial for medical diagnoses, is fraught with imperfections. Let's explore some of these key elements:

In medical testing, two main errors can occur: false positives and false negatives.

• A **false positive** is when a test indicates a person has a disease, but they do not.
• A **false negative** is when a test indicates a person does not have a disease, but they actually do.

These errors can impact the reliability of test results. The test in our exercise reported a 98% chance of being correct if you do have the disease, which means there is a 2% probability of a false negative. Moreover, it is 95% accurate at identifying the lack of a disease, leading to a 5% chance of a false positive.

Understanding these probabilities is key to interpreting any test results accurately and is essential in calculating the real likelihood of being afflicted, as seen in the exercise's use of Bayes' Theorem.

The Basics of Probability Theory Applied to Testing

Probability theory is the mathematical study of uncertainty. It provides a framework to quantify how likely events are to occur, making it invaluable in situations such as disease testing, where outcomes are uncertain.

The basis of probability calculations begins with defining the total chances of certain events (like testing positive), which can then further be explored using techniques like Bayes' Theorem.

In the realm of disease testing:

• **Initial Probabilities:** Determine basic likelihoods like having the disease initially (1% in our exercise), knowing that such diseases might be rare.
• **Combined Event Probabilities:** Use known results, such as testing positive, to reassess the likelihood of having the disease through conditional probability.

By applying probability theory and understanding these concepts, one can better interpret medical test results and navigate the uncertainty inherent in disease diagnosis.