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Binomial Distribution

We come across probability distributions quite frequently in mathematics. There are different types of probability distributions in probability theory. The binomial distribution is a very important study within probability distributions. In a particular case, Binomial distribution is also known as the "Bernoulli distribution". Binomial distribution deals with the experiments which have one of the two types of results either yes or no, either true or false, success or failure. It is a discrete probability distribution. This distribution can be used under the following conditions:

• $n$, the number of trails is finite.

• There are only two possible outcomes in each trial arbitrarily called success and failure.

• The probability of success in each trail is $p$ and is constant for each trail. $q$ = $1 - p$ is then termed as the probability of failure and is constant for each trail. 

• All the "$n$" trails are independent.

The trails satisfying the above four conditions are also known as Bernoulli trails or Binomial distribution.

When $n$ = $1$, the binomial distribution is a Bernoulli distribution.

Related Calculators
Binomial Distribution Calculator Binomial Calculator
Binomial Confidence Interval Calculator Binomial Multiplication Calculator
 

Definition

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A random variable $X$ is defined to have a Binomial distribution if the discrete density function of $X$ is given by

$f_x\ (x)$ = $f_x\ (x,\ n,\ p)$

= $\left\{\begin{matrix} ^nC_xP^x(1 - P)^{n - x} & ;for \ x = 0, 1, ...., n\\  0 & ;otherwise \end{matrix}\right.$

= $C_x^n\ P^x(1\ -\ P)^{(n\ -\ x)}\ I_{(0,1, 2,...,n)}\ (x)$

where, the two parameter $n$ and $p$ satisfies $0 \leq  P\  \leq 1,\ n$ ranges over positive integers. A distribution denoted by the density function is called a binomial distribution.
Formula:

The formula for binomial distribution is as follows,

$\frac{\lambda^{r}}{r!}\ [1\ -\ \frac{1}{n}][1\ -\ \frac{2}{n}]\ ....\ [1\ -\ \frac{r\ -\ 1}{n}]\ \times\ \frac{[(1\ -\ \frac{\lambda}{n})^{\frac{n}{\lambda}}]^{-\ \lambda}}{(1\ -\ \frac{\lambda}{n})^r}$ ... (1)

Let us find the probability of getting "$x$" success in "$n$" trails, where the probability of getting success is "$p$", the probability of getting failure is "$q$". In "$n$" trails the total number of possible ways of getting "$r$" success in $C_r^n$.

Some characteristics of Binomial distributions are:

1) The same action is repeated n times, repetition must be identical.

2) Each trial must be independent of all others.

3) Only $2$ outcomes are possible (a Success or a Failure).

Graph

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Any distribution may be represented in graphic form. Since the binomial distribution tells us the probability of success out of n trials. Therefore place values of $r$ along the horizontal axis and values of $P(r)$ on the vertical axis. A histogram is an appropriate graph of a binomial distribution.

Graph of Binomial distribution for $n$ = $6$ and $P$ = $0.70$, where r varies from $0$ to $6$.

Binomial Distribution Graph

The binomial distribution for a random variable $X$ with parameters $n$ and $p$ represents the sum of $n$ independent variables $Z$ which may assume the values $0$ or $1$. If the probability that each $Z$ variable assumes the value $1$ is equal to $p$, then the mean is equal to $p$. By the addition properties for independent random variables, the mean of the binomial distribution is equal to the sum of the mean of the n independent $Z$ variables, $\mu$ = $np$. Mean of binomial distribution is also called as the expected value of binomial distribution.
Mean of Binomial Distribution:

If $p$ is the probability and $n$ the number of events, then the mean of the $X$ is given by,

$\mu$ = $np$.

Variance and Standard Deviation

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The binomial distribution for a random variable $X$ with parameters $n$ and $p$ represents the sum of $n$ independent variables $Z$ which may assume the values $0$ or $1$. If the probability that each $Z$ variable assumes the value $1$ is equal to $p$, then the variance is equal to $p(1 - p)$. By the addition properties for independent random variables, the variance of the binomial distribution is equal to the sum of the variances of the $n$ independent $Z$ variables,

$\sigma^2$ = $np(1 - p)$

Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$.

Then the variance of $X$ is given by:

$var(X)$ = $np(1 - p)$

Standard Deviation

Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$.

Then, the standard deviation of $X$ is given by:

$\sigma$ = $\sqrt{Var(X)}$

where, $var(X)$ = $np(1 - p)$.

Negative Binomial Distribution

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The probability distribution of a negative binomial random variable is called a negative binomial distribution. The Pascal distribution and Polya distribution are special cases of the negative binomial. Suppose a negative binomial experiment consists of $x$ trials and results in $r$ successes.
If the probability of success on an individual trial is P, then the negative binomial probability is:

$b(X,\ r,\ P)$ = $^{X\ -\ 1}\ C_{r\ -\ 1}\ P^r\ (1\ -\ P)^{X\ -\ r}$


Negative Binomial Distribution Example

Given below are some of the examples on negative binomial distribution.

Solved Example

Question: Victor is a basketball player. He is a 60% free throw shooter. During the season, what is the probability that Victor makes his second free throw on his fourth shot.

Solution:
 
Step 1:
This is an example of a negative binomial experiment. The probability of success (P) is 0.60 (because, Victor is a 60% free throw shooter. That means his probability of making a free throw is 0.60).

And  the number of trials (x) is 4, and the number of successes (r) is 2.

=> P = 0.60, 1 - P = 1 - 0.60 = 0.40

x = 4, r = 2

Step 2:
To solve this problem, we enter these values into the negative binomial formula.

b(X, r, P) = $^{x - 1}C_{r - 1}$ Pr (1 - P)x - r

= $^3C_1$ x (0.6)2 (0.4)4 - 2

 = $^3C_1$ x (0.6)2 (0.4)2

= 3 x 0.36 x 0.16

= 0.1728

Thus, the probability that Victor will make his second successful free throw on his fourth shot   is 0.1728.
 

Table

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Binomial distribution table computes the probability of obtaining $x$ successes in $n$ trials of a binomial experiment with probability of success $P$.

Binomial Distribution Table

Beta Binomial Distribution

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The Beta-Binomial distribution is used to model the number of successes in n binomial trials when the probability of success $p$ is a Beta $(a,\ b)$ random variable. A Beta-Binomial distribution returns a discrete value between $0$ and $n$. The beta-binomial is a compound distribution of the beta and binomial distributions. It is obtained when the parameter $p$ in the binomial distribution is assumed to follow a beta distribution.

A random variable $X$ follows the beta distribution with the probability function

$P(x; n,\ a,\ b)$ = $\begin{pmatrix} n\\x \end{pmatrix}$ $\frac{B(a + x, b + n - x)}{B(a, b)}$, $x$ = $0, 1, 2, ............, n$.

where, $B(a,\ b)$ is the beta function with parameter $a$ and $b$.

Cumulative Binomial Distribution

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The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial. These outcomes are appropriately labeled "success" and "failure". Cumulative Binomial Distribution is Used to find the probability that $x$ is less than or equal to some amount. The formula for the binomial cumulative probability function is

$F(x, p, n)$ = $\sum_{x = 0}^c \begin{pmatrix}n\\x \end{pmatrix}$ $p^x\ (1\ -\ p)^{n\ -\ x}$

Cumulative Binomial Distribution Table


Cumulative Binomial probability distribution table computes the cumulative probability of obtaining $x$ successes in n trials of a binomial experiment with probability of success $p$.

Cumulative Binomial Distribution Table

Poisson Binomial Distribution

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The Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials. Let $X$ be discrete random variable that can take on the values $0, 1, 2$,....... such that the probability function of $X$ is given by

$f(x)$ = $P(X$ = $x)$ = $\frac{\lambda^x e^{-1}}{x!}$, $x$ = $0, 1, 2$,..................

where, $\lambda$ is a positive constant.

A Poisson Binomial distribution is a sum $X$ = $X_1 + X_2 + ............. + X_n$ of '$n$' independent Bernoulli random variables which may have arbitrary expectations.

Examples

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Given below are few binomial distribution examples for your better understanding to calculate binomial distribution.

Solved Examples

Question 1: On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate get 4 or more correct answers just by guessing ?

Solution:
 
Let x is the number of correct answers, then x has binmonial distribution with

n = 5, p = $\frac{1}{3}$ and q = 1 - p = 1 - $\frac{1}{3}$ = $\frac{2}{3}$

Required probability = P(x $\geq$ 4) = P(x = 4) + P(x = 5)

= 5C4 ($\frac{2}{3})^{5 - 4}$ ($\frac{1}{3})^4$ + 5C5 ($\frac{2}{3})^{5-5}$($\frac{1}{3})^5$

= 5 * $\frac{2}{3}$ * $\frac{1}{81}$ + 1 * 1 * $\frac{1}{243}$

= $\frac{10 + 1}{243}$

= $\frac{11}{243}$

 

Question 2: Find the probability of getting 3 exactly twice in 7 throws of a die.
Solution:
 
Probability of getting 3 on throwing a die = $\frac{1}{6}$

=> p = $\frac{1}{6}$ and q = 1 - p = 1 - $\frac{1}{6}$ = $\frac{5}{6}$


Here n = 7, then using binomial distribution,

Required probability = P(x = 2)

= 7C2 ($\frac{5}{6})^{7 - 2}$ ($\frac{1}{6})^2$

= $\frac{7*6}{1*2}$ ($\frac{5}{6})^5$ * $\frac{1}{36}$

= $\frac{7}{12}$ ($\frac{5}{6})^5$

 

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