# APPENDIX A

# Probability

## A.1 INTRODUCTION TO PROBABILITY THEORY

In this section we give a brief introduction to elementary probability theory, which is the basis of the mathematical approach to modeling failures. The presentation is nonrigorous. The objective is to develop an intuitive feel for the topic that forms the foundation for most models used in solving reliability-related problems.

### 1.1 Sample Space and Events

Consider an experiment whose outcome is not known in advance but is such that the set of all outcomes (called the sample space ) is known. Any subset of the sample space is called an event. The concept is illustrated in the following two examples.

*Example 1:* A manufactured part may be in working or failed state because of variations in the quality of manufacturing. In this case there are two possible outcomes and the sample space is the set .

*Example 2:* In the case of an unreliable system, the set *E*_{i} may be used to denote the event that the system will fail at age *t* which lies in the interval [*t*_{i−1}, *t*_{i}) with *t*_{0} = 0 and *t*_{i}, *i* = 1, 2, …, an increasing sequence. In this case, the sample space is the set of intervals .

Events can be viewed as sets in the sample space. Consider ...