Question

The mammogram is helpful for detecting breast cancer in its early stages. However, it is an imperfect diagnostic tool. According to one study, 1186.6 of every 1000 women between the ages of 50 and 59 that do not have cancer are wrongly diagnosed (a "false positive"), while 1.1 of every 1000 women between the ages of 50 and 59 that do have cancer are not diagnosed (a "false negative"). One in 38 women between 50 and 59 will develop breast cancer. If a woman between the ages of 50 and 59 has a positive mammogram, what is the probability that she will have breast cancer?

Short answer

The probability that a woman between the ages of 50 and 59 will have breast cancer given a positive mammogram is approximately 2.1%

Step-by-step solution

Analyze the given data

First, it's important to understand the given data. The question statess that 1 in 38 women between 50 and 59 will develop breast cancer, so the chance of having breast cancer in this age group is \(\frac{1}{38} \approx 0.0263\). If a woman does not have breast cancer, there is a chance of 1-0.0263 = 0.9737. Further, the rate of false negatives is 1.1 in 1000 or \(0.0011\), and the rate of false positives is 1186.6 in 1000 or \(1.1866\). This gives four possibilities: A woman can have cancer and receive a positive test (true positive), have cancer and receive a negative test (false negative), not have cancer and receive a positive test (false positive), or not have cancer and receive a negative test (true negative).

Calculate the Probabilities

Next, multilpy the probabilities of each scenario:TP = P(Cancer) x P(Pos | Cancer) = \(0.0263 x (1 - 0.0011) = 0.02625\)FN = P(Cancer) x P(Neg | Cancer) = \(0.0263 x 0.0011 = 0.000029\)FP = P(No Cancer) x P(Pos | No Cancer) = \(0.9737 x 1.1866 = 1.1554\)TN = P(No Cancer) x P(Neg | No Cancer) = \(0.9737 x (1 - 1.1866) = -0.1816\)

Calculate Probability of Cancer Given a Positive Test (Bayes’ Theorem)

Bayes’ theorem for this scenario would be P(Cancer | Pos) = \(\frac{P(Cancer) x P(Pos | Cancer)}{P(Pos)} \), where P(Pos) is the sum of true positives and false positives, so P(Pos) = TP + FP = 0.02625 + 1.1554 = 1.18164. Hence, P(Cancer | Pos) = \(\frac{0.0275}{1.18164} = 0.021 \) or about 2.1%.