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Question 10.3: Approximately 1% of women have breast cancer (i.e., 99% do n......

Approximately 1% of women have breast cancer (i.e., 99% do not) with 80% of mammogram tests detecting breast cancer when it is there (i.e., 20% miss it), and 9.6% of mammograms detect breast cancer when it is not there (false positive) with 90.4% correctly returning a negative). What is the probability that a woman has breast cancer given a positive mammogram test?

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Let T be the event that the patient’s mammogram test result is positive, and D the event that the tested person has breast cancer. Then the desired probability is P(D|T) and is given by

P(D\mid T)=\frac{P(DT)}{P(T)}=\frac{{P}({T}\mid{D}){P}({D})}{{P}({T}\mid{D}){P}({D})+{P}({T}\mid{D}^{c}){P}({D}^{c})}

= \frac{0.8*0.01}{0.8*0.01+0.096*0.99}

= 7.8%

That is, the probability that she has breast cancer following a positive mammogram test is 0.078.

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