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.

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

The experiment  has n repeated trials.

Each trial can have two possible outcomes. One is success and another one is failure.

Here the trials are independent.

Mean = n * P.
Variance = n * P * (1 – P).
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.

Cumulative Binomial probability

It  refers to the binomial probability falls within a specified range that is greater than or equal to a mentioned lower limit and less than or equal to a mentioned upper limit.

For example

Cumulative binomial probability of obtaining 5 or fewer heads in 10 times of a coin.

b( x <= 5; 10, 0.5)=   b( x = 0; 10, 0.5) + b( x = 1; 10, 0.5) +…… + b ( x = 5; 10, 0.5)