A certain disease has an incidence rate of 0.2%. If the false negative rate is 7% and the false positive rate is 1%, compute the probability that a person who tests positive actually has the disease
To compute the probability that a person who tests positive actually has the disease, we can use Bayes' theorem.
Let's define the following probabilities:
- P(D) = probability of having the disease = 0.2% = 0.002
- P(N) = probability of not having the disease = 100% - P(D) = 99.8% = 0.998
- P(POS|D) = probability of testing positive given that the person has the disease = 100% - false negative rate = 93% = 0.93
- P(POS|N) = probability of testing positive given that the person does not have the disease = false positive rate = 1% = 0.01
Now, let's calculate the probability that a person who tests positive actually has the disease using Bayes' theorem:
P(D|POS) = (P(D) * P(POS|D)) / (P(D) * P(POS|D) + P(N) * P(POS|N))
P(D|POS) = (0.002 * 0.93) / (0.002 * 0.93 + 0.998 * 0.01)
P(D|POS) = 0.00186 / (0.00186 + 0.00998)
P(D|POS) = 0.00186 / 0.01184
P(D|POS) ≈ 0.157
Therefore, the probability that a person who tests positive actually has the disease is approximately 0.157 or 15.7%.