**The Binomial Distribution**

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1 Introduction

The **Binomial Distribution** is used when an event has only two possible outcomes. Within the context of a series of trials these are generally called **success** and **failure**. The probabilities of each of these outcomes occurring are denoted by* p *and *q *respectively. Since the probability of a success is* p, *the probability of a failure is therefore . If an experiment is carried out with trials, the probability of successes (and consequently failures) is given by

where N.B. by definition. Because the binomial distribution was originally discovered by Jacob Bernoulli (1654-1705), it is sometimes called the **Bernouilli **distribution.

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1.1 Examples

i) What is the probability of obtaining 4 heads out of 7 tosses of an unbiased coin?

Solution:

The tossing of a head is classed as a success. Consequently, the probability of 4 heads is given by

ii) What is the probability of dealing 2 spades if 6 cards are dealt from a normal pack of playing cards?

Solution:

The probability of dealing a spade is The probability of success (dealing a spade) and failure (not dealing a spade) are respectively and . Consequently, the probability of dealing 2 spades in 6 cards is given by

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2 Mean, variance and standard deviation of the Binomial Distribution

The mean, , of a binomial distribution is given by .

The variance, , of a binomial distribution is given by .

The standard deviation,, of a binomial distribution is given by .

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2.1 Examples

Find the mean, variance and standard deviation for i) and ii) in the examples given above

Solutions:

i) . Consequently, are given by

,

.

ii) . Consequently, are given by

,

.

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3 Moment generating function of the Binomial Distribution

The moment generating function of the binomial distribution is given by

.

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