Applications - Probability

Question

Answer the following questions using the information given in the table below. The table is based on the assumption that 2% of breast tumors are malignant. This table also assumes that mammogram screening is 80% accurate.




                                           Tumor Is Malignant - Tumor Is Benign - - Total


Positive Mammogram                      640                      7840                8480

Negative Mammogram                    160                      31360               31520

Total                                              800                      39200              40000



A.
Suppose the patient has a positive mammogram. What is the chance that she really does have cancer?

(Round to the nearest tenth of a percent.)



B.
What is the chance of a positive mammogram, given that the patient has cancer?

(Round to the nearest tenth of a percent.)



C.
Suppose a patient has a negative mammogram. What is the chance that she actually has cancer?

(Round to two decimal places.)



D.
What is the chance of a positive mammogram, given that the patient does not have cancer?

(Round to the nearest tenth of a percent.)

Answer 1

A.  8480 women had a positive mammogram, of which 640 actually had cancer.  So with a positive mammogram, the chance of having cancer are 640 / 8480 = 7.5%.

 

B.  800 women who received mammograms had cancer, or which 640 had a positive mammogram.  So if the patient has cancer, the chance of a positive mammogram is 640 / 800 = 80%

 

C.  31520 women had a negative mammogram, of which 160 actually had cancer.  So the chance that a woman with a negative mammogram has cancer is 160 / 31520 = 0.5%

 

D. 39200 women did not have cancer, of which 7840 had a positive mammogram.  So the chance that a woman does not have cancer but still has a positive mammogram is 7840 / 39200 = 20%