A screening test for a disease shows a positive test result in 95% of all cases when the disease is actually present and in 20% of all cases when it is not. When the test was administered to a large number of people, 21.5% of the results were positive. What is the prevalence of the disease?
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To find the prevalence of the disease, we need to understand the concept of conditional probability and use Bayes' theorem.
Let's define the following probabilities:
P(D) represents the prevalence of the disease (the probability that a randomly selected individual has the disease).
P(Pos|D) represents the probability of testing positive given that the disease is present.
P(Pos|~D) represents the probability of testing positive given that the disease is not present.
According to the problem:
P(Pos|D) = 0.95 (the test shows a positive result when the disease is present).
P(Pos|~D) = 0.20 (the test shows a positive result when the disease is not present).
P(Pos) = 0.215 (the overall probability of testing positive).
Using Bayes' theorem, we can calculate the prevalence of the disease as follows:
P(D|Pos) = (P(Pos|D) * P(D)) / P(Pos)
We can substitute the known values into the equation:
P(D|Pos) = (0.95 * P(D)) / 0.215
To find the prevalence of the disease, we need to solve this equation. However, we need additional information to determine the value of P(D).