Washington Mathematics Official Publication of the Washington State Mathematics Council
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An Activity Applying Probability to Random Drug Testing
Brenda Cathcart
Northwest Yeshiva High School
When my students ask me how they are ever going to use math after school, one of my replies is that it is their responsibility as citizens to be well informed about issues. This includes understanding the mathematics behind many social issues and statistics. The following activity is one that addresses that necessity of the general populace to be "numerate".
The Mathematics of Random Drug Testing
In the June 13, 1994 "Dear Abby" in the Seattle Times, there was a letter from a person who was tested for HIV when donating blood. When the test came back positive, the person was terrified but didn't tell anyone. The doctor at the blood bank advised a retest in six months. Instead, the person went to another doctor who said that because no risk factors were present, the positive test was probably a false positive. The second doctor repeated the test and it came back negative. The letter concluded with the warning that false positives are not uncommon and to get retested before worrying.
A false positive is a sample that tests positive for a disease or substance but the disease or substance is not really there. How often can we expect false positives to occur? What is the mathematics involved in drug and disease testing?
Let's consider the HIV situation above and assume the test for HIV is 98% accurate. That means for those who have the disease, it correctly reports the existence of HIV in 98% of those tested (true positive), and fails to report the existence of HIV in 2% of those tested (false negative). For those who do not have HIV, it correctly reports the absence of the disease in 98% of those tested (true negative) and reports the existence of the disease in 2% of those tested(false positive). The incidence of HIV in the United States in 1997 was about .26%. This means that the probability of a person chosen at random has the disease is .0026.
a) Write the appropriate probabilities along each branch of the tree diagram and find the probability of each branch.
b) If 100,000 people are screened for the disease, about how many people can be expected to test positive for the disease? Hint: P(disease and positive test)+ P(no disease and positive test)= P(positive test)
c) Of the people who test positive for the disease, how many people actually have the disease?
d) What is the probability that a person who tests positive for the disease actually has it?
e) Repeat b through d above to find the probability that a person who tests negative for the disease does not have the disease.
f) Write a few sentences about how probability supports or does not support random HIV testing. Can the same thing be said about random drug testing in general? Why or why not?
Possible extensions:
© 1999 WSMC and Richard T. Edgerton