A certain disease has an incidence rate of 0.5%. If the false negative rate is 7% and the false positive rate is 2%, 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 assume that there are 10,000 people in the population.
First, we need to calculate the number of people who have the disease. Given an incidence rate of 0.5%, out of 10,000 people, we have:
Number of people with the disease = 0.5% of 10,000 = 0.005 * 10,000 = 50
The false positive rate is 2%, so the number of people without the disease who test positive would be:
Number of people without the disease but test positive = 2% of 10,000 = 0.02 * 10,000 = 200
The false negative rate is 7%, so the number of people with the disease who test negative would be:
Number of people with the disease but test negative = 7% of 50 = 0.07 * 50 = 3.5
Now, let's calculate the total number of people who test positive:
Number of people who test positive = Number of people with the disease + Number of people without the disease but test positive
Number of people who test positive = 50 + 200 = 250
Finally, the probability that a person who tests positive actually has the disease can be calculated as:
Probability = Number of people with the disease / Number of people who test positive
Probability = 50 / 250 = 0.2
Therefore, the probability that a person who tests positive actually has the disease is 0.2, or 20%.
To compute the probability that a person who tests positive actually has the disease, we need to use Bayes' theorem. Bayes' theorem states that:
P(A|B) = (P(B|A) * P(A)) / P(B)
where:
P(A|B) is the probability of event A occurring given that event B has occurred,
P(A) is the probability of event A occurring independently of event B,
P(B|A) is the probability of event B occurring given that event A has occurred, and
P(B) is the probability of event B occurring independently of event A.
In this case:
Event A is having the disease,
Event B is testing positive.
We are given the following information:
- The incidence rate of the disease is 0.5% or 0.005.
- The false negative rate is 7% or 0.07.
- The false positive rate is 2% or 0.02.
To calculate the probability of actually having the disease given a positive test result (P(A|B)), we need to calculate the probability of testing positive given that the person has the disease (P(B|A)) and the probability of having the disease (P(A)), as well as the probability of testing positive regardless of having the disease (P(B)).
Let's calculate each probability step-by-step:
1. Probability of testing positive given that the person has the disease (P(B|A)):
Since the false negative rate is the probability of testing negative given that the person has the disease, the probability of testing positive given that the person has the disease is 1 - false negative rate:
P(B|A) = 1 - 0.07 = 0.93
2. Probability of having the disease (P(A)):
The incidence rate is the probability of having the disease:
P(A) = 0.005
3. Probability of testing positive (P(B)):
To calculate this, we need to consider two scenarios: Testing positive given that the person has the disease (P(B|A)) and testing positive given that the person does not have the disease (false positive rate):
P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)
P(not A) is the probability of not having the disease, which is equal to 1 - P(A):
P(not A) = 1 - 0.005 = 0.995
P(B) = 0.93 * 0.005 + 0.02 * 0.995
Now we can calculate the probability of actually having the disease given a positive test result (P(A|B)):
P(A|B) = (P(B|A) * P(A)) / P(B)
P(A|B) = (0.93 * 0.005) / (0.93 * 0.005 + 0.02 * 0.995)
Performing the calculation, we get:
P(A|B) ≈ 0.189 or 18.9%
Therefore, the probability that a person who tests positive actually has the disease is approximately 18.9%.