### Introduction

When the ideas of probability are applied to engineering (and many other areas) there are occasions when we need to calculate conditional probabilities other than those already known. For example, if production runs of ball bearings involve say, four machines, we might know the probability that any given machine produces faulty ball bearings. If we are inspecting the total output prior to distribution to users, we might need to know the probability that a faulty ball bearing came from a particular machine. Even though we do not address the area of statistics known as Bayesian Statistics here, it is worth noting that Bayes’ theorem is the basis of this branch of the subject.

#### Prerequisites

- understand the ideas of sets and subsets.
- understand the concepts of probability and events.
- understand the addition and multiplication laws and the concept of conditional probability.

#### Learning Outcomes

- understand the term ‘partition of a sample space’
- understand the special case of Bayes’ theorem arising when a sample space is partitioned by a set and its complement
- be able to apply Bayes’ theorem to solve basic engineering related problems

#### Contents

1 The theorem of total probability1.1 A partition of a sample space

2 Bayes’ theorem

3 Special cases

3.1 The theorem of total probability: special case

3.2 Bayes’ theorem: special case