Homework Help - Statistics!!?

Question

The problem states:
Breast cancer is the most common form of cancer in women, affecting about 10% of women. The chance of breast cancer increases as women age, and it is recommended that women over 40 test for its presence. For the use of the mammogram to detect breast cancer, typical values reported are sensitivity=0.86 and specificity=0.88. The prevalence for women over 40 yrs is 0.01.

(NOTE: sensitivity means person with cancer and tests positive. specificity means person does not have cancer and test negative, and prevalence means that they can haev breast cancer)

a) You are over 40 yrs ld and recived a positive test. What is the probability that you actually have breast cancer?

b) You are over 40 yrs old and received a negative test. What is the probablity that you actually don't have breast cancer?

Answer 1

This is a Bayes' Theorem problem.
Let C = the probability that a woman has cancer = 0.10
Let C- = the probability that a woman does not have cancer = 0.90
Let T = the probability that the test is positive
Let T- = the probability that the test is negative
P(C|T-) is defined as false negative
P(C-|T) is defined as false positive.

Assuming that the prevalence = 0.10

a) Then, by Bayes' Theorem P(C|T) = P(T|C)P(C)/[P(T|C)P(C) + P(T|C-)P(C-]
Evaluating, P(C|T) = (0.86)(0.10)/[(0.86)(0.10) + (0.12)(1 - 0.88)] = 0.856

b) P(C-|T-) = (0.12)(0.10)/[(0.12)(0.10) + (0.88)(1-0.90)] = 0.12

Answer 2

From my reading, for (a) it would be 1-14/86 and for (b) 1- 12/88. .

Answer 3

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