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Chapter 12: Problem 10

It is known that \(33 \%\) of people over the age of 50 around the world have some kind of arthritis. A test has been developed to detect arthritis in individuals. This test was given to a large group of individuals with confirmed cases and a positive test result was achieved in \(87 \%\) of the cases.That same test gave a positive test to \(4 \%\) of individuals that do not have arthritis. If this test is given to an individual at random and it tests positive, what is the probability that the individual has this disease?

Short Answer

Expert verified
The probability is approximately 91.46%.

Step by step solution

01

Identify the Given Information

We know that 33% of people have arthritis, which means the probability of having arthritis ( P(A) ) is 0.33. A test gives a positive result for 87% of people with arthritis, so the probability of a positive test given arthritis ( P(T+|A) ) is 0.87. The test also gives a positive result for 4% of people without arthritis, so the probability of a positive test given no arthritis ( P(T+|A') ) is 0.04.
02

Determine Probabilities of Subgroups

We need to calculate the probability of someone testing positive regardless of whether they have arthritis or not. This is the total probability of a positive test (P(T+)). We find this by:\[P(T+) = P(T+|A) \cdot P(A) + P(T+|A') \cdot P(A')\]where P(A')is the probability of not having arthritis, which is 0.67, since P(A') = 1 - P(A). Substituting the known values gives:\[P(T+) = 0.87 \cdot 0.33 + 0.04 \cdot 0.67\].
03

Calculate the Total Probability of a Positive Test

Compute the above expression:\[P(T+) = 0.87 \cdot 0.33 + 0.04 \cdot 0.67 = 0.2871 + 0.0268 = 0.3139\].
04

Apply Bayes' Theorem

We want the probability that an individual has arthritis given that they tested positive, P(A|T+). Using Bayes' Theorem:\[P(A|T+) = \frac{P(T+|A) \cdot P(A)}{P(T+)}\]. Substitute the known values:\[P(A|T+) = \frac{0.87 \cdot 0.33}{0.3139}\].
05

Compute the Probability Using Bayes' Theorem

Calculate the probability:\[P(A|T+) = \frac{0.2871}{0.3139} \approx 0.9146\].
06

Interpret the Result

The probability that an individual who tests positive actually has arthritis is approximately 91.46%.

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Key Concepts

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

Conditional Probability
Conditional probability is a fundamental concept in statistics and probability theory. It helps us understand the likelihood of an event occurring, given that another event has already occurred. In the context of our exercise, we are interested in determining the probability that a person has arthritis if they test positive for it.
In our example:
  • The probability of having arthritis is denoted as \( P(A) = 0.33 \).
  • The probability of testing positive if a person has arthritis is \( P(T+|A) = 0.87 \).
  • We also see that the probability of testing positive if a person does not have arthritis is \( P(T+|A') = 0.04 \).
These probabilities will help us apply Bayes' Theorem later to find the conditional probability of interest, \( P(A|T+) \). Conditional probability is about updating our belief when additional information is given.
False Positive
A false positive arises in testing scenarios when a test indicates the presence of a condition or disease when it is actually not present. This can often lead to unnecessary stress or further medical procedures.
For arthritis testing:
  • The test erroneously identifies arthritis in people without it in 4% of cases, as shown by \( P(T+|A') = 0.04 \).
  • This means there is a 4% chance of receiving a false positive result.
False positives highlight the importance of understanding the accuracy and reliability of medical tests. Knowing the probability of false positives assists in better interpreting test results and making informed decisions.
Probability Theory
Probability theory is the mathematical framework that allows us to analyze random events and assess outcomes' likelihoods. This theory underpins our understanding of conditional probabilities and helps in examining how likely it is that certain events will occur under uncertain circumstances.
  • It provides tools like Bayes' Theorem to calculate probabilities effectively.
  • Bayes' Theorem combines information from prior beliefs (such as 33% prevalence of arthritis) with current data (test results) to update our understanding.
In our case, probability theory models the entire testing process, allowing us to determine the likelihood of someone having arthritis based on a positive test result. This is represented mathematically as \( P(A|T+) \). It is a powerful tool for decision-making in uncertain environments.
Tests for Accuracy
Tests for accuracy are crucial when evaluating medical diagnostics. They provide insight into how often a test correctly identifies a condition when present (sensitivity) and how often it correctly identifies a lack of a condition when absent (specificity).
  • The sensitivity of our arthritis test is characterized by \( P(T+|A) = 0.87 \), meaning it correctly identifies 87% of those with the condition.
  • The specificity can be inferred indirectly, given \( P(T-|A') \). Since \( P(T+|A') = 0.04 \), the test correctly identifies 96% as negative when the disease is truly absent, representing the specificity.
Having tests with high sensitivity and specificity is critical in the medical field to reduce false positives and false negatives. This ensures more reliable test outcomes, leading to better patient care and resource allocation.

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