Question 16

A certain disease has an incidence rate of $0.3 \%$. If the false negative rate is $4 \%$ and the false positive rate is $4 \%$, compute the probability that a person who tests positive actually has the disease.

Give your answer accurate to at least 3 decimal places
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Step 1 ：Define the given values: The incidence rate of the disease, P(A), is 0.003. The probability of testing positive given that the person has the disease, P(B|A), is 0.96 (1 minus the false negative rate). The false positive rate, P(B|~A), is 0.04. The probability of not having the disease, P(~A), is 0.997 (1 minus P(A)).

Step 2 ：Calculate the total probability of testing positive, P(B), using the formula P(B) = P(B|A) * P(A) + P(B|~A) * P(~A). Substituting the given values, we get P(B) = 0.96 * 0.003 + 0.04 * 0.997 = 0.04276.

Step 3 ：Use Bayes' theorem to find the probability that a person has the disease given that they tested positive, P(A|B). The formula is P(A|B) = P(B|A) * P(A) / P(B). Substituting the given values, we get P(A|B) = 0.96 * 0.003 / 0.04276 = 0.06735266604303086.

Step 4 ：Round the final answer to at least 3 decimal places. The probability that a person who tests positive actually has the disease is approximately \(\boxed{0.067}\).