Chapter 6, Problem 100CR

False positive results are not uncommon with mammograms, a test used to screen for breast cancer. For a woman who has a positive mammogram, the probability that she actually has breast cancer is less than 0.05 if she is under 40 years old, and ranges from 0.05 to 0.109 if she is over 40 years old (“Breast Cancer Screenings: Does the Evidence Support the Recommendations?,” Significance [August 2016]: 24–37). If a woman with a positive mammogram is selected at random, are the two events

B = event that selected woman has breast cancer and

A = event that selected woman is over 40 years old

independent events? Justify your answer using the given information.

To determine

Check whether the events A and B are independent.

Explanation of Solution

The given information is about the mammogram test. The probability that a woman under 40 years old has breast cancer is 0.05 and the probability of a woman over 40 years old has breast cancer is ranges from 0.05 to 0.109.

Event A denotes the events that the selected woman is over 40 years old and B denotes the events that the selected woman has breast cancer.

The probability that a woman under 40 years old has breast cancer is 0.05 and the corresponding probability for woman over 40 years old ranges from 0.05 to 0.109.

Thus, the probability of breast cancer changed with woman under and over 40 years old.

Hence, the events A and B are not independent.