A certain disease has an incidence rate of 0.3%. If the false negative rate is 6% and the false positive rate is 2%, compute the probability that a person who tests positive actually has the disease.

Solve using a probability tree:

(sick)/0.003*0.9=0.00282 (positive)
/0.003\0.003*0.06=0.00018 (false negative)
1
\0.997/0.997*0.02=0.01994 (false positive)
(not \0.997*0.98=.97706 (negative)
sick)

check: .00282+.00018+0.01994+.97706=1.0000 √

"the probability that a person who tests positive actually has the disease."
Looking at the tree diagram,
the probability of a person testing positive is 0.00282+0.01994=0.02276.
Probability of a person testing positive AND having the disease is 0.00282.

Therefore the conditional probability of a person having the disease GIVEN that he tests positive is
P(D|+)=P(D∩+)/P(+)
=0.00282/0.02276
where
D=event that the person has the disease
+=event that the person tests positive.

To determine the probability that a person who tests positive actually has the disease, we can use Bayes' theorem. Let's denote the following events:

A = Person has the disease
B = Person tests positive

We want to find P(A | B), 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)

where P(B | A) is the probability of testing positive given that the person has the disease, P(A) is the probability of having the disease, and P(B) is the probability of testing positive.

Given information:
- Incidence rate = 0.3% = 0.003 (P(A))
- False negative rate = 6% = 0.06 (P(B | A'))
- False positive rate = 2% = 0.02 (P(B' | A'))

First, let's calculate P(B), the probability of testing positive:
P(B) = P(B | A) * P(A) + P(B | A') * P(A')

To calculate P(B), we also need to find P(A'), the probability of not having the disease:
P(A') = 1 - P(A)

P(A') = 1 - 0.003
P(A') = 0.997

Next, we can calculate P(B):

P(B) = P(B | A) * P(A) + P(B | A') * P(A')
P(B) = 0.02 * 0.003 + 0.06 * 0.997
P(B) ≈ 0.00006 + 0.05982
P(B) ≈ 0.05988

Now, we can substitute the values into Bayes' theorem to find P(A | B):

P(A | B) = (P(B | A) * P(A)) / P(B)
P(A | B) = (0.02 * 0.003) / 0.05988
P(A | B) ≈ 0.00006 / 0.05988
P(A | B) ≈ 0.001002

Therefore, the probability that a person who tests positive actually has the disease is approximately 0.001002, or about 0.1002%.

To compute the probability that a person who tests positive actually has the disease, we need to use Bayes' theorem. Bayes' theorem relates conditional probabilities and can be stated as follows:

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, event A is having the disease, and event B is testing positive. We are asked to compute the probability of having the disease given that the test result is positive.

Let's break down the information given:

- The incidence rate of the disease is 0.3%. This means that the probability of randomly selecting a person who has the disease is 0.003, or 0.3%.
- The false negative rate is 6%, which means that the probability of testing negative when actually having the disease is 0.06, or 6%.
- The false positive rate is 2%, which means that the probability of testing positive when not having the disease is 0.02, or 2%.

Now, let's use Bayes' theorem to calculate the probability of having the disease given a positive test result:

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

P(Positive|Disease) is the probability of testing positive given that a person has the disease. In this case, it is 1 minus the false negative rate, or 1 - 0.06 = 0.94.

P(Disease) is the probability of having the disease, which is 0.003.

P(Positive) is the probability of testing positive, which is calculated by summing up the probabilities of testing positive when having the disease (true positive) and testing positive when not having the disease (false positive).

P(Positive) = P(Positive|Disease) * P(Disease) + P(Positive|Not Disease) * P(Not Disease)
= 0.94 * 0.003 + 0.02 * (1 - 0.003)

Now we can substitute these values into the equation:

P(Disease|Positive) = (0.94 * 0.003) / (0.94 * 0.003 + 0.02 * (1 - 0.003))

Calculating this expression will give us the probability that a person who tests positive actually has the disease.