## Binomial Distribution

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# Binomial Distribution

The Binominal Distribution also known as ‘Bernoulli Distribution’ is associated with the name of a Swiss Mathematician Jacob Bernoulli (1654-1705). Binomial distribution is probability distribution expressing the probability of one set of dichotomous alternatives, i.e., success or failure.This distribution has been used to describe a wide variety of processes in business and the social sciences as well as other areas. The type of process which gives rise to this distribution is usually referred to as Bernoulli trial or as Bernoulli process. The mathematical model for a Bernoulli process is developed under specific set of assumptions involving the concept of a series of experimental trials. These assumptions are:

1. An experiment is performed under the same conditions for a fixed number of trials, say, n.

2. In each trial, there are only two possible outcomes of the experiment “success” or “failure”. Stated in somewhat different language, the sample space of possible outcomes on each experimental trial is: S = {failure, success}

3. The probability of a success denoted by p remains constant from trial to trial. The probability of a failure denoted by q is equal to (1-p). If the probability of success is not the same in each trial we will not have binomial distribution. For example, if 5 balls are drawn at random from an urn containing 10 white and 20 red balls, this is a binomial experiment if each ball is replaced before another is drawn. If the balls are drawn without replacement, the probability of drawing white ball changes each time a ball is taken from the item and we no longer have a binomial experiment.

4. The trials are statistically independent, i.e., the outcomes of any trial or sequence of trails do not affect the outcomes on subsequent trials.

This model is useful to answer questions such as this: If we conduct an experiment n times under the stated conditions, what is the probability of obtaining exactly x success? More specifically, suppose 10 coins are tossed together. What is the probability of obtaining exactly two heads?

How binomial distribution arises can be seen from the following:

If a coin is tossed once there are two outcomes, namely tail or head. The probability obtaining a head or p = ½ and the probability of obtaining a tail or q = 1/2. Thus (q + p) = 1. These are terms of the binomial (q + p).

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