Question

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) Write down what is given in the problem, both explicitly and implicitly, using the events defined in the problem description.

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

c) Interpret your answer for P(C|+).

Answer 1

Solution:-

Given That:-

One and a 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

(a)It is given that,

From the given data,

1.5% of women have breast cancer.

92% of women having breast cancer tests positive.

9% of women have false positive mammograms i.e. 9% of women not having breast cancer tests positive.

(b)

The event that a randomly selected woman has breast cancer given that her mammogram is positive is denoted as (C|+).

Using Bayes' theorem we have,

(c)

The value of P(C|+) interprets that 13.47% of women who are tested positive in mammogram have breast cancer.