Question

The National Cancer Institute estimates that 3.65% of women in their 60s get breast cancer. A mammogram can typically identify correctly 85% of cancer cases and 95% of cases without cancer. What is the probability that a woman in her 60s who has a positive test actually has breast cancer?

Answer 1

39.17% probability that a woman in her 60s who has a positive test actually has breast cancer

Explanation:

Bayes Theorem:

Two events, A and B.

`P(B|A) = (P(B)*P(A|B))/(P(A))`

In which P(B|A) is the probability of B happening when A has happened and P(A|B) is the probability of A happening when B has happened.

In this question:

Event A: Positive test.

Event B: Having breast cancer.

3.65% of women in their 60s get breast cancer

This means that
`P(B) = 0.0365`

A mammogram can typically identify correctly 85% of cancer cases

This means that
`P(A|B) = 0.85`

Probability of a positive test.

85% of 3.65% and 100-95 = 5% of 100-3.65 = 96.35%. So

`P(A) = 0.85*0.0365 + 0.05*0.9635 = 0.0792`

What is the probability that a woman in her 60s who has a positive test actually has breast cancer?

`P(B|A) = (0.0365*0.85)/(0.0792) = 0.3917`

39.17% probability that a woman in her 60s who has a positive test actually has breast cancer