Download Conditional Probability and Independence

Conditional Probability and Independence
Worksheet 7
1. Human Chorionic Gonadotropin (hCG) pregnancy tests or home pregnancy tests determine pregnancy
through the detection of the hormone hCG in a woman’s urine. Tests too soon after conception can
result in false negatives and certain aspects of recent health history of a women can result in false
positives. Let A be the event tests positive and C be the event woman is pregnant.
(a) Given a false positive rate of 3% and a false negative rate of 5%, find P (A|C) and P (Ac |C).
(b) Find the probability that a woman tests positive if her assessment that the probability that she
is pregnant is 60%.
(c) Find the probability that a women is pregnant given that she tests positive.
(d) Determine this conditional probability if her assessment that the probability that she is pregnant
is p. Show the results in a plot of p versus the probability that a women is pregnant.
2. Athletes in high level competitions like the Olympics are subject to urine drug tests (UDTs) for the use
of banned substances. False positives can occur from in vivo metabolic conversions of a legal substance,
exposure to nonillicit sources, or laboratory error. False negative UDT results include limited assay
specificity, absence of the substance in the urine, or laboratory error. Let A be the event tests positive
and C be the event athlete used a banned substance.
(a) Given a false positive rate of 1% and a false negative rate of 0.5%, find P (A|C) and P (A|C c ).
(b) Find the probability that an athlete tests positive if the fraction of athletes that use a banned
substance is 2%.
(c) Find the probability that an athlete used the drug given that the test is positive.
(d) What would have a bigger impact on this probability, reduction in false positive probability or
false negative probabiity? Explain your answer.
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