Question

Suppose that a screening test for breast cancer has 95% sensitivity and 90% specificity. Assume 1% of the population being screened truly has breast cancer.

a. If you really do have breast cancer, what is the probability that the test says you do?

b. If you really do not have breast cancer, what is the probability that the test says you do?

c. The screening test is applied to a total of 15 people; 5 who really do have cancer and 10 who do not. What is the probability that it gets all of the screens correct? (That is, the five people with cancer get a positive result on the screen and the ten without cancer get a negative result).

d. You just tested positive. What is the probability that you really do have breast cancer, given your test result?

e. An alternative screening test is introduced with 93% sensitivity and 92% specificity. Which test has a better Positive Predictive Value? Negative Predictive Value?

Answer 1

Solution :-

From the given data ... Let's define

Event B = Person is having Breast Cancer actually. => ~B = Person is not having Breast Cancer

Event T = Test says person is having Breast Cancer => ~T = Test says person is not having Breast Cancer

Given

Sensitivity

P(T/B) = 0.95

Specificity

P(~T/~B) = 0.90

P(B) = 0.01

a) P(T/B) = ?

P(T/B) = 0.95

b) P(T/~B) = ?

P(T/~B) = 1 - P(~T/~B) = 1 - 0.90 = 0.1

c) Probability that it gets all of the screens correct

d) P(B/T) = ?

Using Bayes theorem :-

e) Now for alternative screening test given

P(T/B) = 0.93 and

P(~T/~B) = 0.92

So now , previous test has a better Positive Predictive Value than alternative test. And this alternative test has a better Negative Predictive Value than previous