Question

Suppose the probability of a false positive result on a mammogram is 7% and that radiologists' interpretations of mammograms are mutually independent in the sense that whether or not a radiologist finds a positive result on one mammogram does not influence whether or not a radiologist finds a positive result on another mammogram. Assume that a woman has one mammogram every year for ten years.

(a) What is the probability (as a %) that she will have no false positive results during that time?
(b) What is the probability (as a %) that she will have at least one false positive result during that time?
(c) What is the probability (as a %) that she will have exactly two false positive results during that time?
(d) Suppose that the probability of a false negative result on a mammogram is 3%, and assume that the probability that a randomly chosen woman has breast cancer is 0.0003.
(i) If a woman has a positive test result one year, what is the probability (as a %) that she actually has breast cancer?
(ii) If a woman has a negative test result one year, what is the probability (as a %) that she actually has breast cancer?

Answer 1

a) 48.4%

b) 51.6%

c) 12.3%

d) i) 93.0% ii) 3.0%

Explanation:

a)

P(fp) = probability of false positive result

P(fp) = 0.07

P(fp = 0) = (1 - 0.07)¹⁰

P(fp = 0) = 0.4839... ⇒ 48.4%

b)

P(fp = 1) = 1 - 0.4839...

P(fp = 1) = 0.516... ⇒ 51.6%

c)

P(fp = 2) = 10C2(0.07)²(0.93)⁸

P(fp = 2) = 0.123... ⇒ 12.3%

d)

P(fn) = probability of false negative

P(b) = probability a randomly selected woman has breast cancer

P(fn) = 0.03

P(b) = 0.0003

If she has a positive test, it is either a true positive, meaning she actually has breast cancer, or a false positive, meaning she doesn't;

If P(fp) = 0.07, then the probability of a true positive is 0.93 or 93%

ii)

If she has a negative test, a true negative means she doesn't have breast cancer, and a false negative means she does have breast cancer;

The probability of a false negative is 0.03 or 3%

(I'm not entirely sure about part d, but I've answered as I've understood it according to my logic, they seem to be somewhat trick questions unless I've incorrectly understood the questions)