Application of Bayes' Theorem in Medical Testing

Consider a medical test designed to identify a specific disease. The test has a true positive rate (sensitivity) of 90% and a true negative rate (specificity) of 85%. In the general population, the prevalence of the disease is 2%. A patient received a positive result from the test. What is the probability that the patient actually has the disease?

To solve this problem, you will need to use Bayes' Theorem, which is given by the formula:

$$P(Disease | Positive) = \frac{P(Positive | Disease) \times P(Disease)}{P(Positive)}$$

Where:

• $$P(Disease | Positive)$$ is the probability of having the disease given a positive test result.
• $$P(Positive | Disease)$$ is the true positive rate of the test.
• $$P(Disease)$$ is the prior probability of having the disease.
• $$P(Positive)$$ is the total probability of a positive test result.

First, calculate the necessary probabilities:

• $$P(Positive | Disease) = 0.90$$ (sensitivity)
• $$P(Disease) = 0.02$$ (prevalence)
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• To find $$P(Positive)$$, consider both true positives and false positives, where:
• $$P(Positive) = P(Positive | Disease) \times P(Disease) + P(Positive | No Disease) \times P(No Disease)$$
• $$P(No Disease) = 1 - P(Disease) = 0.98$$
• $$P(Positive | No Disease) = 1 - P(Negative | No Disease) = 1 - 0.85 = 0.15$$ (false positive rate)