Question

The following hypothetical scenario concerns a population of women aged 50 and older. We know that 1% of this population has breast cancer at this time. The probability of testing positive on a mammogram is 90% for women who actually do have cancer (true positive result), and 8% for women who do not have it (false positive result). (a) Find the proportion of all negative mammogram results for this population. (8 points) (b) Given that a randomly selected woman has a negative mammogram result, what is the probability that she has cancer? (10 points) (c) How do you interpret your results for part (b)? (2 points)

Answer 1

Let A denote the event of having Cancer and A' be denote the event not having Cancer then,

P(A) = 0.01

P(A') = 0.99

Also, let X denote the event of testing positive and Y denote the event of testing negative

Therefore, P(X|A) = 0.90 and P(X|A') = 0.08

and  P(Y|A) = 0.10 and P(Y|A') = 0.92

Then,

a) proportion of all negative mammogram is

P(Y) = P(A) * P(Y|A) + P(A') * P(Y|A')

= 0.01*0.1 + 0.99 * 0.92

= 0.9118

b) P(A|Y) = P(Y|A) *P(A) / ( P(Y|A') *P(A') + P(Y|A) *P(A))

= (0.1 * 0.01) / ( (0.99 * 0.92) + (0.1 * 0.01) )

  ​​​​​​

c) Around 0.1% of the people who have tested negative will have cancer.