Question

Suppose the probability of a false positive result on a mammogram is 4%. Suppose that the probability of a false negative result on a mammogram is 2%. Assume that the probability that a randomly chosen woman has breast cancer is 0.0002. If a woman has a positive test result, what is the probability that she actually has breast cancer

Answer 1

0.0049 = 0.49% probability that she actually has breast cancer

Explanation:

Conditional Probability

We use the conditional probability formula to solve this question. It is

`P(B|A) = (P(A \cap B))/(P(A))`

In which

P(B|A) is the probability of event B happening, given that A happened.

`P(A \cap B)` is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question:

Event A: Positive test result

Event B: Breast cancer

Probability of a positive test result:

4% of (1 - 0.0002)

98% of 0.0002

So

`P(A) = 0.04*(1 - 0.0002) + 0.98*0.0002 = 0.040188`

Having breast cancer and testing positive.

98% of 0.0002

So

`P(A \cap B) = 0.98*0.0002 = 0.000196`

What is the probability that she actually has breast cancer?

`P(B|A) = (0.000196)/(0.040188) = 0.0049`

0.0049 = 0.49% probability that she actually has breast cancer