An athletic woman in her twenties arrives at the emergency department complaining of dizziness after running in hot weather. An electrocardiogram is used to check for a heart attack and the patient generates an abnormal result. The test has a false positive rate 0.1 (the probability of an abnormal result when the patient is normal) and a false negative rate of 0.1 (the probability of a normal result when the patient is abnormal). Furthermore, it might be assumed that the prior probability of a heart attack for this patient is 0.001. Although the abnormal test is a concern, you might be surprised to learn that the probability of a heart attack given the electrocardiogram result is still less than 0.01. See “Why Clinicians are Natural Bayesians” (2005, bmj.com) for details of this example and others.
The key is to properly combine the given probabilities. Furthermore, the exact same analysis used for this medical example can be applied to tests of engineered products. Consequently knowledge of how to manipulate probabilities in order to assess risks and make better decisions is important throughout scientific and engineering disciplines. In this chapter the laws of probability are presented and used to assess risks in cases such as this one and numerous others.