### Home > INT3 > Chapter 6 > Lesson 6.3.1 > Problem6-81

6-81.

Any system that is responsible for detecting relatively rare events is going to have problems with false positives. You saw this in problem 6‑72 regarding testing for HIV. Consider cases including such things 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 that is responsible for sounding an alarm is $96$% accurate. If the alarm is sounding, what is the probability that event A has not occurred (false alarm)?

1. Make a model for this situation.

Make a two-way table or tree diagram.
$P(\text{no Event A given yes alarm sounds})=\frac{\text{no Event A and yes alarm sounds}}{\text{yes alarm sounds}}$