Day 53: Reversing the Condition
Reversing the condition in probability can be a challenging concept for students in AP Stats. An example of a situation where reversing the condition is necessary is:
Athletes and Drug Testing Over 10,000 athletes competed in the 2008 Olympic Games in Beijing. The International Olympic Committee wanted to ensure that the competition was as fair as possible. So the committee administered more than 5000 drug tests to athletes. All medal winners were tested, as well as other randomly selected competitors. Suppose that 2% of athletes had actually taken (banned) drugs. No drug test is perfect. Sometimes the test says that an athlete took drugs, but the athlete actually didn’t. We call this a false positive result. Other times, the drug test says an athlete is “clean,” but the athlete actually took drugs. This is called a false negative result. Suppose that the testing procedure used at the Olympics has a false positive rate of 1% and a false negative rate of 0.5%. What’s the probability that an athlete who tests positive actually took drugs?
I have tried many different ways of helping students including introducing Bayes Theorem as a way to help students navigate these kinds of problems. I think the underlying issue is that my students come into class thinking they understand conditional probability when in fact they really don’t (Type I error?). One wonderful activity is Abby’s Kennels from the NCTM’s Navigating through Probability in Grades 9-12. This activity gives students a concrete experience with what a conditional probability really means. I regularly use this activity with my precalculus class if we study probability. In fact, many of my AP Stats students were in my precalc class when we studied probability and did the Abby’s Kennel activity. They still had a strong intuitive sense of what a conditional probability is – yay!!
It can be daunting for students to wade through verbiage like that above, let alone organize the information in an understandable way. I have found that giving my students the “anatomy of a tree diagram” has really helped! In particular, helping them see how to find the probability of the second event when they don’t have explicit information about it. Here is an example of the diagram I draw with them:
I have also re-created the “anatomy of a two-way table” although I personally don’t find tables as easy to set up when given conditional probabilities. This year’s students are more ready to create a tree diagram. From it, we can develop the reversed condition.