Assume that the probability of breast cancer equals .04 for women in the 50-59 a

Question

Assume that the probability of breast cancer equals .04 for women in the 50-59 age group. Furthermore, if a women does have breast cancer, the probability of a true positive mammogram (correct detection of breast cancer) equals .80 and the probability of a false negative mammogram (a miss) equals .20. On the other hand, if a women does not have breast cancer, the probability of a true negative mammogram (correct nondetection) equals .90 and the probability of a false positive
mammogram (a false alarm) equals .10. (Hint: Use a frequency analysis to answer questions. To facilitate checking your answers with those in the book, begin with a total of 1,000 women, then branch into the number of women who do or do not have breast cancer, and finally, under each of these numbers, branch into the number of women with positive and negative mammograms.)

(a) What is the probability that a randomly selected woman will have a positive mammogram? .
(b) What is the probability of having breast cancer, given a positive mammogram?
(c) What is the probability of not having breast cancer, given a negative mammogram?

Explanation / Answer

Ans:

a)P(cancer)=0.04

P(no cancer)=1-0.04=0.96

P(+ve/cancer)=0.8

P(-ve/cancer)=1-0.8=0.2

P(-ve/no cancer)=0.9

P(+ve/no cancer)=1-0.9=0.1

a)P(+ve)=P(+ve/cancer)*P(cancer)+P(+ve/no cancer)*P(no cancer)

=0.8*0.04+0.1*0.96

=0.032+0.096

=0.128

b)P(cancer/+ve)=0.8*0.04/0.128=0.032/0.128=0.25

c)P(-ve)=1-0.128=0.872

P(not cancer/-ve)=P(-ve/not cancer)*P(not cancer)/P(-ve)=0.9*0.96/0.872=0.864/0.872=0.991

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