A certain disease has an incidence rate of 0.6%. If the false negative rate is 5% 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 define the following:

P(D) = probability of having the disease = 0.6%
P(~D) = probability of not having the disease = 100% - P(D) = 99.4%
P(+|D) = probability of testing positive given that the person has the disease (sensitivity) = 100% - false negative rate = 95%
P(+|~D) = probability of testing positive given that the person does not have the disease (specificity) = 1% - false positive rate = 98%

Now, we can calculate the probability of having the disease given a positive test using Bayes' Theorem:

P(D|+) = (P(D) * P(+|D)) / ((P(D) * P(+|D)) + (P(~D) * P(+|~D)))

Substituting the values:

P(D|+) = (0.006 * 0.95) / ((0.006 * 0.95) + (0.994 * 0.02))

Calculating this expression:

P(D|+) = 0.0057 / (0.0057 + 0.01988)
P(D|+) = 0.0057 / 0.02578
P(D|+) ≈ 0.221

Therefore, the probability that a person who tests positive actually has the disease is approximately 22.1%.

To compute the probability that a person who tests positive actually has the disease, we can use Bayes' theorem.

Let's break down the problem step by step:

1. Define the given probabilities:
- Incidence rate: 0.6% = 0.006 (this represents the proportion of people in the population who have the disease).
- False negative rate: 5% = 0.05 (this is the probability that a person who has the disease tests negative).
- False positive rate: 2% = 0.02 (this is the probability that a person who does not have the disease tests positive).

2. Calculate the following probabilities:
- True positive rate (sensitivity): This is the complement of the false negative rate. In this case, it is 1 - 0.05 = 0.95.
- True negative rate (specificity): This is the complement of the false positive rate. In this case, it is 1 - 0.02 = 0.98.

3. Apply Bayes' theorem:
The formula for Bayes' theorem is:
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(B|A) is the probability of event B occurring given that event A has occurred.
- P(A) is the probability of event A occurring.
- P(B) is the probability of event B occurring.

In this case, we want to find P(Disease|Positive), where:
- A = Disease
- B = Positive

P(Disease|Positive) = [P(Positive|Disease) * P(Disease)] / P(Positive)

4. Calculate the probabilities:
- P(Positive|Disease) is the sensitivity (0.95).
- P(Disease) is the incidence rate (0.006).
- P(Positive) can be calculated by considering both true positive and false positive cases:
P(Positive) = [P(Positive|Disease) * P(Disease)] + [P(Positive|No Disease) * P(No Disease)]
P(Positive) = (0.95 * 0.006) + (0.02 * (1 - 0.006))

5. Substitute the values into the formula and solve for P(Disease|Positive).

Here's the final equation:
P(Disease|Positive) = (0.95 * 0.006) / [(0.95 * 0.006) + (0.02 * (1 - 0.006))]

By plugging in the values into the equation, you can calculate the probability that a person who tests positive actually has the disease.