The Binomial Distribution

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 Binomial_files\Binomial_MathML_0.jpg. If an experiment is carried out with Binomial_files\Binomial_MathML_1.jpg trials, the probability of Binomial_files\Binomial_MathML_2.jpg successes (and consequently Binomial_files\Binomial_MathML_3.jpg failures) is given by


where Binomial_files\Binomial_MathML_5.jpg N.B. Binomial_files\Binomial_MathML_6.jpg by definition. Because the binomial distribution was originally discovered by Jacob Bernoulli (1654-1705), it is sometimes called the Bernouilli distribution.

1.1  Examples

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


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?


The probability of dealing a spade is Binomial_files\Binomial_MathML_8.jpgThe probability of success (dealing a spade) and failure (not dealing a spade) are respectively Binomial_files\Binomial_MathML_9.jpgand Binomial_files\Binomial_MathML_10.jpg. Consequently, the probability of dealing 2 spades in 6 cards is given by


2  Mean, variance and standard deviation of the Binomial Distribution

The mean, Binomial_files\Binomial_MathML_12.jpg, of a binomial distribution is given by Binomial_files\Binomial_MathML_13.jpg.

The variance, Binomial_files\Binomial_MathML_14.jpg, of a binomial distribution is given by Binomial_files\Binomial_MathML_15.jpg.

The standard deviation,Binomial_files\Binomial_MathML_16.jpg, of a binomial distribution is given by Binomial_files\Binomial_MathML_17.jpg.

2.1  Examples

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


i) Binomial_files\Binomial_MathML_18.jpg. Consequently, Binomial_files\Binomial_MathML_19.jpg are given by

Binomial_files\Binomial_MathML_20.jpg ,



ii) Binomial_files\Binomial_MathML_23.jpg. Consequently, Binomial_files\Binomial_MathML_24.jpg are given by

Binomial_files\Binomial_MathML_25.jpg ,



3  Moment generating function of the Binomial Distribution

The moment generating function of the binomial distribution is given by