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.
Solution:
Step 1: identify the simple events, and represent each with its own symbol:
A person either has cancer or does not have cancer. Therefore, these two events are mutually exclusive and partition the set of living individuals.
A1 = The patient has cancer.
A2 = The patient does not have cancer.
The problem also discusses the event of a positive test result:
B = The test says the patient has cancer.
Step 2: Label the probabilities provided in terms of these established symbols (letters):
P(A1) = 0.0001
P(B/A1) = 0.9
P(B/A2) = 0.001
Step 3: Using the information in the question and the events defined above, we know that we want to find P(A1/B).
Step 4: 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)]
Step 5: The problem has given us a value for every term in the above formula expect P(A2). However, we know that A1 and A2 are complements of each other since the only cancerous states in a given person are the presence of cancer or the absence of cancer. Therefore,
P(A2) = 1 - P(A1).
P(A2) = 1 - 0.0001
P(A2) = 0.9999
Step 6: Plugging in all of our values, we get
P(A1/B) = [(0.9)(0.0001)] / [(0.9)(0.0001) + (0.001)(0.9999)
P(A1/B) = 0.00009 / 0.0010899
P(A1/B) = 0.08257638, which is approximately 8%
Therefore, the probability that a patient has cancer given that the test says she has cancer is about 0.0826, or about 8%.
So, for every 100 people who test positive for cancer using this new procedure, only about 8 people actually are positive for cancer.
Step 1: identify the simple events, and represent each with its own symbol:
A person either has cancer or does not have cancer. Therefore, these two events are mutually exclusive and partition the set of living individuals.
A1 = The patient has cancer.
A2 = The patient does not have cancer.
The problem also discusses the event of a positive test result:
B = The test says the patient has cancer.
Step 2: Label the probabilities provided in terms of these established symbols (letters):
P(A1) = 0.0001
P(B/A1) = 0.9
P(B/A2) = 0.001
Step 3: Using the information in the question and the events defined above, we know that we want to find P(A1/B).
Step 4: 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)]
Step 5: The problem has given us a value for every term in the above formula expect P(A2). However, we know that A1 and A2 are complements of each other since the only cancerous states in a given person are the presence of cancer or the absence of cancer. Therefore,
P(A2) = 1 - P(A1).
P(A2) = 1 - 0.0001
P(A2) = 0.9999
Step 6: Plugging in all of our values, we get
P(A1/B) = [(0.9)(0.0001)] / [(0.9)(0.0001) + (0.001)(0.9999)
P(A1/B) = 0.00009 / 0.0010899
P(A1/B) = 0.08257638, which is approximately 8%
Therefore, the probability that a patient has cancer given that the test says she has cancer is about 0.0826, or about 8%.
So, for every 100 people who test positive for cancer using this new procedure, only about 8 people actually are positive for cancer.