What is the probability?

Question

1.    One and a half percent (1.5%) of women over 55 have breast cancer. Ninety-two percent (92%) of all women who have breast cancer test positive on mammograms. Nine percent (9%) of women will have false positive mammograms. Define the events: C: {a woman has breast cancer} and +: {the mammogram is positive}. 

a)    If a randomly selected woman’s mammogram is positive, what is the probability that she has breast cancer?

Answer 1

Hi Jiwoo, your probability question can be solved by the application of Baye's Theorem:

P(A|B) = P(B|A)P(A) / [ P(B|A)P(A) + P(B|A^c)P(A^c) ]

where we assume A has a perfectly complementary event A^c - that is A is a binary event. In the case where A is not binary and doesn't have a perfectly complementary event, then denominator becomes a summation term: ∑P(B|A_i)P(A_i)

For your problem, if we translate the statement into conditional probability:

P(C|+) <--- applying Baye's theorem

= P(+|C)P(C) / [ P(+|C)P(C) + P(+|C^c)P(C^c) ]

here C^c: {a woman over age 55 who doesn't have breast cancer}

P(+|C) = Prob(mammogram is + GIVEN a woman over age 55 has breast cancer) = 0.92

P(+|C^c) = Prob(mammogram is + GIVEN a woman over age 55 doesn't have breast cancer) = 0.09 <-- False positive event is when a healthy woman is wrongly diagnosed to have breast cancer via the mammogram test

P(C) = Prob(a woman over age 55 has breast cancer) = 0.015

P(C^c) = Prob(a woman over age 55 who doesn't have breast cancer) = 1 - 0.015 = 0.985

So,

P(C|+) = 0.92*0.015 / (0.92*0.015 + 0.09*0.985) ≈ 0.135

The result is telling us that given that a positive mammogram, the probability a randomly selected woman over the age of 55 has breast cancer is 0.135. This seems to imply mammogram is not "accurate". This counter-intuitive result is actually due to the fact that majority (98.5%) of all women over the age of 55 don't have breast cancer and the somewhat high false positive rate of the mammogram test is wrongly diagnosing healthy women with breast cancer. If the false positive rate of the mammogram becomes very small, say 0.1%, then P(C|+) ≈ 0.933. So the moral of the problem is that medical test should have very low false positive rate in order for it to be accurate.