Suppose that a medical test has a 92% chance of detecting a disease if a person has it, and a 94% chance of correctly indicating that the disease is absent if the person really does not have the disease. Suppose 10% of the population has the disease:
a)what is the probability that a randomly chosen person will test positive?
b)Suppose a randomly chosen person does test positive. What is the probability that this person really has the disease?

Question

Suppose that a medical test has a 92% chance of detecting a disease if a person has it, and a 94% chance of correctly indicating that the disease is absent if the person really does not have the disease. Suppose 10% of the population has the disease: 
a)what is the probability that a randomly chosen person will test positive?
b)Suppose a randomly chosen  person does test positive. What is the probability that this person really has the disease?

phi · Answer

a)what is the probability that a randomly chosen person will test positive?

you can get a positive result 2 ways:   
1)  the person has the disease and the test says YES
2)  the person does not have the disease and the test says YES

the chance for case 1)  is the chance the person has the disease (10% according to the question) times the probability the test works correctly (92% chance of correctly indicating)
so 0.1*0.92= 0.092

for case 2)  the chance the person does not have the disease times the chance the test gives a false positive.  Can you do this case?

anonymous · Answer

I understand, but I don't know what numbers to plug in for the second case.

phi · Answer

for case 2)
what is the chance the person does not have the disease?

phi · Answer

I would say 1 - chance a person has the disease

anonymous · Answer

wait, you lost me.

phi · Answer

we are doing case 2
2)  the person does not have the disease and the test says YES

we are told
Suppose 10% of the population has the disease: 

what is the chance a person does not have the disease?

anonymous · Answer

90%?

phi · Answer

yes.  if 10% have the disease then 90% do not.  or  1-0.10= 0.90

we are also told 
 94% chance of correctly indicating that the disease is absent if the person really does not have the disease

from that statement, what is the chance the test gives the wrong answer and says YES, we have got a diseased person.

anonymous · Answer

6%?

phi · Answer

yes.  if it gives the correct answer 94% of the time, what is it doing the remaining 6% of the time?  It is screwing up.

case 2)   the person does not have the disease and the test says YES
use the numbers:  0.90* 0.06=  0.054

add the two cases to get the chance a random person will test positive for the disease.

anonymous · Answer

ok thank you!

phi · Answer

for part b
b)Suppose a randomly chosen  person does test positive. What is the probability that this person really has the disease?

take the ratio of case 1 divided by the sum of case 1 and case 2

anonymous · Answer

ok got it thanks so much!