Mammograms are a relatively inexpensive and nonintrusive way to test for breast

Question

Mammograms are a relatively inexpensive and nonintrusive way to test for breast cancer. Suppose that mammogram screenings are 93% specific and 87% sensitive This means that those without breast cancer test negative 93% of the time and those with breast cancer test positive 87% of the time. Answer the following questions as if 12.6% of all women have breast cancer. Note: These statistics are taken from the Susan G. Komen Foundation webpage at https://ww5.komen. org/BreastCancer/ AccuracyofMammograms.html. (a) What percentage of women receive a false negative result. That is, given that a women tests negative, what is the probability that she has breast cancer? (b) What percentage of women receive a false positive result. That is, given that a women tests positive, what is the probability that she does not have breast cancer?

Explanation / Answer

Mammographs are 93% specific and 87% sensitive .

A:Presence of breast cancer, AN: Absence of breast cancer , B1 : Test positive ,B2 : Test negative

P[AN|B2]=P[Do not have breast cancer | they test negative]=0.93

Similarly P[A|B2]=P[Have breast cancer given they test negative]=0.07=P[A,B2]/P[B2]

P[A|B1 ]=P[Have breast cancer|test postitive ]=0.87

As P[AN|B1]=P[Do not have breast cancer |Test postiive]=1-0.87=0.13

P[A]=Probability that a woman has breast cancer =0.126

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