In a population of 500,000 individuals, a rare disease affects 1 in every 1000 people. A diagnostic test for the disease has been developed, with a sensitivity of 98% and a specificity of 95%. If 100 individuals test positive for the disease, what is the probability that at least 90 of them actually have the disease?
0.01
0.001
0.0001
0.1
To find the probability that at least 90 individuals out of the 100 who test positive actually have the disease, we need to use Bayes' theorem. Let's define the events:
A: Person has the disease
B: Person tests positive for the disease
We are given:
We want to find for at least 90 individuals. We can calculate this using Bayes' theorem:
Using the formula for from Bayes' theorem:
Given :
Substituting the values into the equation, we can calculate to be approximately 0.0001 .
Therefore, the probability that at least 90 out of 100 individuals who test positive actually have the disease is 0.0001.
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