Two Probability Trees A
- Dr. Jack L. Jackson II
Two Probability Trees - Bayesian Techniques
In this problem we have two different, but related, probability trees and the corresponding two-way contingency table. A and B are complementary events and C and D are complementary events. We are given three probabilities from these trees (one straight marginal probability and two conditional probabilities). From this information we can use the structure of probability trees to completely fill in all of the probabilities from both trees and the table. From there we can answer any probability question about this situation. In the app, change the values of the three given probabilities via the sliders or the input boxes. Try to figure out the next step. Slide the step slider one step to the right to check your work.
A Medical Application
One practical application of this problem is a medical situation. Suppose that there is a bad, but rare disease. Let A represent having the disease and thus B is not having the disease. Set A to a small number such as 0.01 = 1%. Suppose there is a test for the disease. Let C be getting a positive test result, and then D is getting a negative test result. The probability of a true positive is the probability that someone with the disease has a positive result: P(C|A). Set P(C|A) to a large number such as 0.99. The probability of a false positive is the probability that someone without the disease will test negative: P(C|B). Set P(C|B) to something low such as 0.04. The main question that we want to know is the following. Suppose that a person goes in for a routine physical exam, showing no symptoms of the disease. He is given the test for this disease and tests positive. What is the probability that he actually has the disease? What does your intuition tell you about the size of this probability? Work through the trees to determine this probability (P(A|C)). If you are the nurse or physician what do you tell the patient? Is the result surprising or not? Notice that P(A), P(C), P(A|C), P(C|A), and P(C & A) are all in at least one of the trees above, but these are typically five different values, possibly very different values.
The situation in this example is a direct illustration of Bayes' Theorem. In Bayes' Theorem we are given a partition of the sample space (in this case into A and B). We know the probabilities of each of the events in this partition. We are also given another event C that intersects at least some of the events in the partition (e.g. here we were given P(A) and used it to find P(B)). We also know the probabilities of C given each element of the partition (e.g. here we know P(C|A) and P(C|B)). We use the techniques above to find each the conditional probabilities in the opposite direction (e.g. here we find P(A|C) and P(B|C)). For my students, I suggest that rather than memorizing Bayes' Theorem, always construct the two probability trees and use your knowledge of the structure of probability trees to fill out both trees. This approach has the added benefit of allowing us to solve many different two tree example that don't fit Bayes' Theorem exactly. For example, in this two tree set up there are many different combinations of three given probabilities that would allow you to completely fill out both trees. See Dr. Jackson's Two Probability Trees B activity for such an example.