A certain disease has an incidence rate of 0.7%. 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.
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To compute the probability that a person who tests positive actually has the disease, we can use Bayes' theorem.
First, let's define some terms:
- P(D) represents the probability of having the disease. In this case, it is given by the incidence rate, which is 0.7% or 0.007.
- P(N) represents the probability of not having the disease, which is equal to 1 - P(D).
- P(Pos|D) represents the probability of testing positive given that a person has the disease. This is equal to 1 - the false negative rate, which is 1 - 0.07 or 0.93.
- P(Neg|N) represents the probability of testing negative given that a person does not have the disease. This is equal to 1 - the false positive rate, which is 1 - 0.01 or 0.99.
Now we can apply Bayes' theorem:
P(D|Pos) = (P(Pos|D) * P(D)) / P(Pos)
To calculate P(Pos), we need to consider two scenarios:
1. A person has the disease and tests positive.
2. A person does not have the disease but still tests positive.
P(Pos) = P(Pos|D) * P(D) + P(Pos|N) * P(N)
Given that a person tests positive, we want to find the probability that person has the disease, which is P(D|Pos).
Now, let's calculate P(D|Pos) step by step:
P(Pos) = P(Pos|D) * P(D) + P(Pos|N) * P(N)
= 0.93 * 0.007 + 0.01 * (1 - 0.007)
Now we can calculate P(D|Pos):
P(D|Pos) = (P(Pos|D) * P(D)) / P(Pos)
= (0.93 * 0.007) / (0.93 * 0.007 + 0.01 * (1 - 0.007))
Calculating this expression, we find that the probability that a person who tests positive actually has the disease is approximately 0.411 or 41.1%.