Homework 4

due Friday, April 26

A.  p. 211, 11,13,14,17,21,24,31,34

B. Bayes Theorem
Bayes Theorem is a means for using P(B | A) to compute P(A | B) for events A and B.  The theorem, in its simplest form,  is this:
P(A | B) = [P(B|A) * P(A) ] divided by     [P(B| A)*P(A) + P(B | not A) * P( not A)]

a) Use Venn diagrams and the rules of probability to show that P(B) = P(A intersect B) UNION P(not A intersection B)
where "not A" is the complement of A.
b) Use the definition of conditional probability, P(A | B) = P(A AND B)/P(B), and the result in (a) to derive Bayes formula.
c) A rare disease occurs in only 4% of the population.  A diagnostic test has been invented to detect the presence of this disease. Of course the test is not perfect.  If you actually have the disease, the test will return a "positive" 99% of the time.  If you do not have the disease, however, the test will still return a positive 5% of the time.  A randomly selected person goes to his physician and is tested for the disease.  The test returns "positive".  What is the probability this person actually has the disease?

Hint: A = "person has disease", B  = "test returns a positive"

Does this probability seem surprisingly low to you?  It might be alarming to think that you could go to the doctor and be diagnosed with a disease when in fact it is unlikely you have the disease. But this problem is a little artificial.  Doctors do not (we hope) randomly give tests to patients.  Instead, the tests are given to those who show symptoms or for some other reason are likely to have the disease.  But this does say something about the success of a mandatory, random drug-testing program at, say, a high school.