Question

A certain disease has an incidence rate of 0.7%. If the false negative rate is 6% and the false positive rate is 1%, compute the probability that a person who tests positive actually has the disease. Give your answer accurate to at least 3 decimal places

Answer 1

To find the probability that a person who tests positive actually has the disease, you can use Bayes' theorem. The probability that a person has the disease is given as the incidence rate, and the false positive and false negative rates are also given. Using these values and Bayes' theorem, you can calculate the desired probability which we get 0.414.

Step-by-step explanation:

To find the probability that a person who tests positive actually has the disease, we can use Bayes' theorem. Let's denote D as the event that a person has the disease and T as the event that a person tests positive. The probability that a person has the disease is given as the incidence rate, which is 0.7% or 0.007. The false positive rate is 1%, so the probability of a false positive is 0.01. The false negative rate is 6%, so the probability of a false negative is 0.06.

Now, using Bayes' theorem:

P(D|T) = (P(T|D) * P(D)) / ((P(T|D) * P(D)) + (P(T|not D) * P(not D)))

Substituting the given values:

P(D|T) = (1 * 0.007) / ((1 * 0.007) + (0.01 * 0.993))

Simplifying the equation:

P(D|T) = 0.007 / (0.007 + 0.00993)

= 0.007 / 0.01693

≈ 0.4137

Therefore, the probability that a person who tests positive actually has the disease is approximately 0.414.