Question

We are given this information:

In a routine intensive screening, 12 of 1000 forty year old randomly selected women were determined to have breast cancer.
Of the 12 who actually had breast cancer, a common test came out positive for 11 of the 12.
Of the 988 without breast cancer, the common test showed positive for 20 of them.

A woman tests positive for breast cancer and the doctor calls her in for intensive interviews.
After the interviews, the doctor determines that this woman falls into a population in which 40% have breast cancer.
The question is: Estimate the probability that this woman actually has breast cancer.

(The answer is not 0.4 or 0.0117)

Answer 1

let A denote teh event that the women diagoned for breast cancer is tested positive

E denote the event that the women diagoned with breast cancer is actually having breast cancer

we have to find:

P()

P(E)=

P(E')=P(women doesnt actually breast cancer)=0.988

P()=P(women tested positive that women was actually having breast cancer)====0.916

P()=P(women tested positive that women was not actually having breast cancer)====0.0202

Now, using bayes theorem:

P()=

P()=

P()=

P()=

P()=0.3551

please rate my answer and comment for doubts.