The Problem: Analysis of Cancer Screening Procedure Accuracy
A patient undergoes an experimental procedure meant to allow early identification of a particular type of cancer. Her results indicate that she does, in fact, have cancer. However, since it is a novel procedure, the test's accuracy is not yet trusted. Before revealing the results to the patient then, her oncologist must find the probability that the patient has cancer given that the test says she has cancer. The only data collected about this procedure is as follows:
The prevalence of the type of cancer identified by the test is 0.0001.
The probability of a positive test result given the patient actually has cancer is 0.9.
The probability of a false positive test result is 0.001.
The prevalence of the type of cancer identified by the test is 0.0001.
The probability of a positive test result given the patient actually has cancer is 0.9.
The probability of a false positive test result is 0.001.
Hint 3:
Since we want to find the probability of the occurrence of a specific earlier event, A1, given the known occurrence of a specific later event, B, we will use Baye's Theorem to solve this problem.
The precedent is the actual presence of cancer in a patient, (event A1).
The known consequence is a test result that indicates the presence of cancer in a patient, (event B).
Baye's Theorem:
P(A1/B) = [P(B/A1) x P(A1)] / [(P(B/A1) x P(A1) + P(B/A2) x P(A2)]
Also, note that we immediately have values for every term in the above formula except P(A2) because we defined the following in Hint 1 based on the information given in the problem:
A1 = The patient has cancer.
A2 = The patient does not have cancer.
B = The test says the patient has cancer.
P(A1) = 0.0001
P(B/A1) = 0.9
P(B/A2) = 0.001
Since we want to find the probability of the occurrence of a specific earlier event, A1, given the known occurrence of a specific later event, B, we will use Baye's Theorem to solve this problem.
The precedent is the actual presence of cancer in a patient, (event A1).
The known consequence is a test result that indicates the presence of cancer in a patient, (event B).
Baye's Theorem:
P(A1/B) = [P(B/A1) x P(A1)] / [(P(B/A1) x P(A1) + P(B/A2) x P(A2)]
Also, note that we immediately have values for every term in the above formula except P(A2) because we defined the following in Hint 1 based on the information given in the problem:
A1 = The patient has cancer.
A2 = The patient does not have cancer.
B = The test says the patient has cancer.
P(A1) = 0.0001
P(B/A1) = 0.9
P(B/A2) = 0.001