Question

Of all 40-year-old women, 1% have breast cancer. If a woman has breast cancer, a mammogram will give a positive indication for cancer 90% of the time. If a woman does not have breast cancer, a mammogram will give a positive indication for cancer 9% of the time. If a 40-year-old mammogram gives a positive indication for cancer, what is the probability that she has cancer?

Answer 1

Answer:

9.17%

Step-by-step explanation:

The probability of a positive indication by a mammogram is given by the probability of positive indication when a woman has cancer, added to the probability of positive indication when a woman does not have cancer.

$P(+) = P(C)*P(+_C)+(1-P(C))*P(+_n)\\P(+) = 0.01*0.9+(1-0.01)*0.09\\P(+) = 0.0981$

Given that the indication is positive, the probability that a woman actually has cancer is:

$P(C|+) = \frac{P(C)*P(+_C)}{P(+)}\\ P(C|+) =\frac{0.01*0.9}{0.0981}\\P(C|+)= 0.0917 = 9.17\%$

The probability that the randomly selected woman has cancer is 9.17%