Binomial Distribution

Binomial Distribution:

Introduction to Binomial Distribution:

The general meaning of a Binomial Distribution is if two mutually exclusive possible outcomes are available means binomial distribution happen.Discrete probability distribution concept is used..It sequence of success in numbers.In binomial distribution,two outcomes are referred.One outcome is success and another one is failure.Probability f success in trials with probability of success on single trial is determined by binomial distribution.p is denote the binomial distribution.


Binomial Distribution is a statistical experiment which means the number of successes in n repeated trials of a binomial experiment. It is also called as Bernoulli distribution or Bernoulli trial.To understand any concept an example is required,now let us look at an example of Binomial Distribution.

For example:

For a clinical trial, a patient may live or die. Here the researcher faces the number of survivors and not how much time the patient lives after treatment.


Properties and Formula for binomial distribution:


For example:

For a clinical trial, a patient may live or die. Here the researcher faces the number of survivors and not how much time the patient lives after treatment.

We take a coin and flipped two times. Here we calculate the count of number of heads(successes). So the binomial distribution is

Number of heads Probability

No head 0.25

One head 0.5

Two head 0.25


Properties of Binomial Distribution:

1. The experiment has n repeated trials.
2. Each trial can have two possible outcomes. One is success and another one is failure.
3. Here the trials are independent.
4. Mean = n * P.
5. Variance = n * P * (1 – P).
6. Standard Deviation = sqrt[ n * P * ( 1 – P ) ].

Binomial distribution Formula:

b(x; n, P) = nCx * Px * (1 - P)n – x

Here the Notation are,

B(x; n, P) = Binomial Probability.

X = successes

N = number of trials

P = Probability of success

nCx = Number of combinations of n trials, x is success.


Example Problem(the binomial distribution):

A die is tossed 6 times. What is the Probability of getting exactly 2 fours?

Solution

Here n = 6, x = 2, probability of success on a single trial = 1/ 6 or 01.167.

Therefore, The binomial probability is,

b( 2; 6, 0.167 ) = 6C2 * ( 0.167 )2 * ( 1 – 0.167)6 – 2

= ( 6! / 2! * (6-2)!) * 0.0279 * ( 0.833)4

= (6! / 2! * 4!) * 0.0279 * 0.481

= 15 * 0.0279 * 0.481

b( 2; 6, 0.167 ) = 0.201. Answer.


Hope you like the above example of Binomial Distribution.Please leave your comments, if you have any doubts.