Any system that is responsible for detecting relatively rare events is going to have problems with false positives. You saw this in problem 11-86 regarding testing for HIV. Consider other cases such as burglar alarms, smoke detectors, red light cameras and drug testing for athletes. All of these systems have proven accuracy and all have a persistent problem of false positives (false alarms).

Consider a hypothetical situation. Suppose that rare event A occurs with a frequency of $\frac{1}{1000}$. Suppose that a detection system for event A responsible for sounding an alarm is $96\%$ accurate. The question is: if the alarm is sounding, what is the probability that event A has not occurred (false alarm)?

1. Make a model for this situation.

   Hint (a):

   Make a two-way table or tree diagram.
   Your model should have labels:
   Event A: yes or no
   Alarm sounds: yes or no

2. If the alarm has been activated, what is the probability that it is false?

   Hint (b):

   $\frac{\text{Event A no, but alarm sounds yes}}{\text{Alarm sounds yes}}$

3. Are the test results mathematically independent of whether event A occurs or not? How could you check this? Explain.

   Hint (c):

   Does the accuracy of the alarm sounding depend on Event A occurring?