A certain disease has an incidence rate of 0.8%. If the false negative rate is 7% and the false positive rate is 3%, compute the probability that a person who tests positive actually has the disease. Show all work.
look at 100,000 people
800 have it
... 56 will show negative
... 744 will show positive
99200 don't have it
... 2976 will show positive
probability of accurate positive
... 744 / (744 + 2976)
To compute the probability that a person who tests positive actually has the disease, we can use Bayes' theorem.
Let's define the following terms:
- P(D) represents the probability of having the disease (prevalence rate) = 0.008 (0.8% converted to decimal).
- P(¬D) represents the probability of not having the disease = 1 - P(D) = 0.992 (1 - 0.008).
- P(T|D) represents the probability of testing positive given that a person has the disease (true positive rate) = 1 - false negative rate = 0.93 (1 - 0.07).
- P(T|¬D) represents the probability of testing positive given that a person does not have the disease (false positive rate) = 0.03 (3%).
Now we can apply Bayes' theorem:
P(D|T) = (P(T|D) * P(D)) / P(T)
P(T) can be calculated using the Law of Total Probability:
P(T) = P(T|D) * P(D) + P(T|¬D) * P(¬D)
P(T) = (0.93 * 0.008) + (0.03 * 0.992)
Simplifying, we find:
P(T) ≈ 0.00744 + 0.02976
P(T) ≈ 0.0372
Now we can calculate P(D|T) using the formula:
P(D|T) = (P(T|D) * P(D)) / P(T)
P(D|T) = (0.93 * 0.008) / 0.0372
Simplifying, we find:
P(D|T) ≈ 0.00744 / 0.0372
P(D|T) ≈ 0.2
Therefore, the probability that a person who tests positive actually has the disease is approximately 0.2, or 20%.
To compute the probability that a person who tests positive actually has the disease, we can use Bayes' theorem. Bayes' theorem is a mathematical formula that relates the conditional probabilities of two events.
Let's define the events:
A: The person has the disease (true positive)
B: The person tests positive
We want to find P(A|B), which is the probability that a person has the disease given that they test positive. According to Bayes' theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
P(B|A) is the probability of testing positive given that the person has the disease and can be calculated using the false negative rate: P(B|A) = 1 - false negative rate = 1 - 0.07 = 0.93
P(A) is the incidence rate of the disease, given as 0.8% or 0.008
P(B) is the overall probability of testing positive and can be calculated using the false positive rate:
P(B) = (P(B|A) * P(A)) + (P(B|not A) * P(not A))
P(B|not A) is the probability of testing positive given that the person does not have the disease and can be calculated using the false positive rate: P(B|not A) = false positive rate = 0.03
P(not A) is the complement of P(A), which is 1 - P(A) = 1 - 0.008 = 0.992
Now, let's calculate P(B):
P(B) = (0.93 * 0.008) + (0.03 * 0.992) = 0.00744 + 0.02976 = 0.0372
Finally, we can substitute these values into Bayes' theorem:
P(A|B) = (0.93 * 0.008) / 0.0372 ≈ 0.198
Therefore, the probability that a person who tests positive actually has the disease is approximately 0.198 or 19.8%.