### Introduction

A situation in which an experiment (or trial) is repeated a fixed number of times can be modelled, under certain assumptions, by the binomial distribution. Within each trial we focus attention on a particular outcome. If the outcome occurs we label this as a success. The binomial distribution allows us to calculate the probability of observing a certain number of successes in a given number of trials.

You should note that the term ‘success’ (and by implication ‘failure’) are simply labels and as such might be misleading. For example counting the number of defective items produced by a machine might be thought of as counting successes if you are looking for defective items! Trials with two possible outcomes are often used as the building blocks of random experiments and can be useful to engineers. Two examples are:

1. A particular mobile phone link is known to transmit 6% of ‘bits’ of information in error. As an engineer you might need to know the probability that two bits out of the next ten transmitted are in error.
2. A machine is known to produce, on average, 2% defective components. As an engineer you might need to know the probability that 3 items are defective in the next 20 produced.

The binomial distribution will help you to answer such questions.

#### Prerequisites

• understand the concepts of probability

#### Learning Outcomes

• recognise and use the formula for binomial probabilities
• state the assumptions on which the binomial model is based

1.1 Notation