Question

{4 marks} Here, we will quickly investigate the importance of understanding conditional prob- abilities when talking with medical patients. This problem is based on a true investigation by Hoffrage and Gigerenzer in 1996. The investigators asked practicing physicians to consider the following scenario:

“The probability that a randomly chosen woman age 40-50 has breast cancer is 1%. If a woman has breast cancer, the probability that she will have a positive mammogram is 80%. However, if a woman does not have breast cancer, the probability she will have a positive mammogram is 10%. Imagine that you are consulted by a woman, age 40-50, who has a positive mammogram but no other symptoms. What is the probability that she actually has breast cancer?”

Twenty-four physicians were asked to respond. The average probability estimate was 70%. Using your knowledge of Bayes’ Rule, determine if the physicians were close in their estimate. Comment on where the error in their judgement may have occurred, and why this may cause problems in their practice.

Answer 1

Given:

P(breast cancer) = P(C) = 0.01

P(not breast cancer) = P(C') = 1 - 0.01 = 0.99

P(positive mammogram | breast cancer) = P(P | C) = 0.80

P(positive mammogram | not breast cancer) = P(P | C') = 0.10

Now,

Using the Bayes's theorem, such that P(A|B) = P(B|A)*P(A)/P(B)

The probability that the woman actually has breast cancer given she has a positive mammogram = 0.07477. Hence there is approximately a 7.477% chance that the women with a positive mammogram. has breast cancer.

Now, the probability can also be obtained by dividing the favorable outcome by total outcomes if the frequency is given using the following formula,

Where the favorable outcomes are the number of women who have breast cancer and a positive mammogram and the total outcomes are the total women with breast cancer.

Now, the physicians overestimate the probability because of the high probability of a positive mammogram when the woman has breast cancer. They fail to correctly describe the total outcome (which should be the number of women who has positive mammogram given breast cancer and the number of women who has positive mammogram given not breast cancer).