Now the incidence rate equals the number of people who where at risk for the disease and got the disease divided by the number of people at risk to get the disease = .006
False negative rate = if it is known someone has the disease what is the probability they have a negative test = people with the disease and a negative test divided by all people who have the disease = .08
False positive rate = if it is known someone does not have the disease what is the probability they have a positive test = people without the disease and a positive test divided by all the people who do not have the disease = .05
False negative rate and false positive rate are known as conditional probabilities. That is it is the probability of something occurring when you already know some information.
Now to determine the probability of someone with a positive test actually having the disease (this would be the positive predictive value) we need to figure out the probability of someone having the disease and testing positive along with the probability of someone testing positive (without knowing their disease status).
Let
T11 = number of people with a positive test and has the disease
T12 = number of people with a positive test and does not have the disease
T21 = number of people with a negative test and has the disease
T22 = number of people with a negative test and does not have the disease
T11+T12 = T+ = number of people with positive test
T11+T21 = TD = number of people with the disease
T12+T22 = TDc = number of people without the disease
T21+T22 = T- = number of people with a negative test
and n = T11+T12+T21+T22 = total number of people at risk for disease
Now the incidence rate equals the number of people who where at risk for the disease and got the disease divided by the number of people at risk to get the disease = .006 = TD/n
False negative rate = if it is known someone has the disease what is the probability they have a negative test = people with the disease and a negative test divided by all people who have the disease = .08 = T21/(T11+T21) = T21/TD
False positive rate = if it is known someone does not have the disease what is the probability they have a positive test = people without the disease and a positive test divided by all the people who do not have the disease = .05 = T12/(T12+T22) = T12/TDc
What the question is asking for is PPV = T11/(T11+T12) = T11/T+ = when it is known someone has a positive test what is the probability they have the disease.
lets say our value for n = 100000 (you can pick any number and the answer will come out correct)
Now the incidence rate equals the number of people who where at risk for the disease and got the disease divided by the number of people at risk to get the disease = .006 = TD/n, if n = 100000 people then .006 = TD/100000, then 100000*.006 = TD = 600 people have the disease of the 100000 people
False negative rate = people with the disease and a negative test divided by all people who have the disease = .08 = T21/(T11+T21) = T21/TD
= T21/6 =.08
= .08*600 = T21 = 48 people have a negative test and have the disease
TD = T11+T21, so that means TD-T21=T11 = 600-48=552 have the disease and have a positive test
n = TD+TDc, so n-TD = TDc = 100000-600 = 99400 people do not have the disease
Then .05 = T12/TDc = T12/99400, so
99400*.05 = T12 = 4970 have a positive test and do not have the disease
Now the question has asked us if it is known that someone has a positive test what is the probability someone has the disease
= T11/(T11+T12)
= 552/(552+4970)
= 0.09996
Thus if it is known someone has a positive test there is a 9.996% chance they have the disease.
The following below is the pure math way to answer this question without using a made up value for n:
if we do (T21/TD)*(TD/n) = T21/n = 0.08*0.006 = 0.00048
Since (T21/n)+(T11/n) = TD/n, we can figure out T11/n = (TD/n)-(T21/n)
= 0.006-0.00048 = 0.00552
We can use the same relationship with the false positive rate to figure out T12 and T22;
1-(TD/n) = TDc/n = 1-0.006 = 0.994
(TD12/TDc)*(TDc/n) = TD12/n = 0.05*0.994 = 0.0497
(TD12/n)+(TD22/n) = TDc/n so it follows that (TDc/n)-(TD12/n)=TD22/n = 0.994-0.0497=0.9443
to now we know that
T11/n = 0.00552
T21/n = 0.00048
T12/n = 0.0497
T22/n = 0.9443
to check out work and make sure the values are correct if we do (T11/n)+(T21/n)+(T12/n)+(T22/n) it should equal to 1, if you do 0.00552+0.00048+0.0497+0.9443 = 1 so our numbers are correct.
Now we can answer the question, which is if it is known someone has a positive test what is the probability they have the disease, now this would be (T11/T+) = (T11/n)/((T11/n)+(T12/n))
= (T11/n)/((T11+YT12)/n)
= T11/(T11+T12)
= T11/T+
So our final answer would be
0.00552/(0.00552+0.0497)
= 0.00552/0.05522
= 0.09996
which means that if someone has a positive test there is a 9.996% chance they have the disease.
