A medical test is available to determine whether a patient has a certain disease. To determine the accuracy of the test, a total of 10,100 people are tested. Only 100 of these people have the disease, while the other 10,000 are disease free. Of the disease-free people, 9800 get a negative result, and 200 get a positive result. The 100 people with the disease all get positive results. Use this information as you answer the questions below. A. Find the probability that the test gives the correct result for a person who does not have the disease. B. Find the probability that the test gives the correct result for a person who has the disease. C. Given that a person gets a positive result, what is the probability that the person actually has the disease?
Answers
Answer:
Step-by-step explanation:
Problems like this are complex and the guide to solve them is to always think of all pairs of possibilities; These are 4 now, namely (infected, positive result), (infected, negative result), (not infected, positive result), (not infected, negative result).
If your test shows positive, then either you are infected and the test is correct or you are not and the test is incorrect. The chance for the first event is 10%*95% by the multiplicative law of probabilities, or 0.095. The chance for the second one (to test positively while not infected) is: (1-0.97)* 0.90=0.027
This is the probability that a person is not infected, times the probability that the test indicates that he is infected. So, the total chance of these two scenarios is 0.027+0.095=0.122. The chance that a person with a positive result is actually positive is thus 0.095/0.122=77.9%. Hence, far from sure.
Similarly as above, the false positive is the complimentary probability. The probability across the population that you test positive while not having the disease is 0.027. Thus, the probability is 0.027/0.122=22.1%. This can be also calculated by 100%-77.9%=22,1%, so this serves as a test of our correctness.