In this post, we’ll discuss Binomial Random Variables.
Prerequisite : Random Variables
A specific type of discrete random variable that counts how often a particular event occurs in a fixed number of tries or trials.
For a variable to be a binomial random variable, ALL of the following conditions must be met:
- There are a fixed number of trials (a fixed sample size).
- On each trial, the event of interest either occurs or does not.
- The probability of occurrence (or not) is the same on each trial.
- Trials are independent of one another.
Mathematical Notations
n = number of trials p = probability of success in each trial k = number of success in n trials
Now we try to find out the probability of k success in n trials.
Here the probability of success in each trial is p independent of other trials.
So we first choose k trials in which there will be a success and in rest n-k trials there will be a failure. Number of ways to do so is
Since all n events are independent, hence the probability of k success in n trials is equivalent to multiplication of probability for each trial.
Here its k success and n-k failures, So probability for each way to achieve k success and n-k failure is
Hence final probability is
(number of ways to achieve k success and n-k failures) * (probability for each way to achieve k success and n-k failure)
Then Binomial Random Variable Probability is given by:
Let X be a binomial random variable with the number of trials n and probability of success in each trial be p.
Expected number of success is given by
E[X] = np
Variance of number of success is given by
Var[X] = np(1-p)
Example 1 : Consider a random experiment in which a biased coin (probability of head = 1/3) is thrown for 10 times. Find the probability that the number of heads appearing will be 5.
Solution :
Let X be binomial random variable with n = 10 and p = 1/3 P(X=5) = ?
Here is the implementation for the same
// C++ program to compute Binomial Probability #include <iostream> #include <cmath> using namespace std;
// function to calculate nCr i.e., number of // ways to choose r out of n objects int nCr( int n, int r)
{ // Since nCr is same as nC(n-r)
// To decrease number of iterations
if (r > n / 2)
r = n - r;
int answer = 1;
for ( int i = 1; i <= r; i++) {
answer *= (n - r + i);
answer /= i;
}
return answer;
} // function to calculate binomial r.v. probability float binomialProbability( int n, int k, float p)
{ return nCr(n, k) * pow (p, k) *
pow (1 - p, n - k);
} // Driver code int main()
{ int n = 10;
int k = 5;
float p = 1.0 / 3;
float probability = binomialProbability(n, k, p);
cout << "Probability of " << k;
cout << " heads when a coin is tossed " << n;
cout << " times where probability of each head is " << p << endl;
cout << " is = " << probability << endl;
} |
// Java program to compute Binomial Probability import java.util.*;
class GFG
{ // function to calculate nCr i.e., number of
// ways to choose r out of n objects
static int nCr( int n, int r)
{
// Since nCr is same as nC(n-r)
// To decrease number of iterations
if (r > n / 2 )
r = n - r;
int answer = 1 ;
for ( int i = 1 ; i <= r; i++) {
answer *= (n - r + i);
answer /= i;
}
return answer;
}
// function to calculate binomial r.v. probability
static float binomialProbability( int n, int k, float p)
{
return nCr(n, k) * ( float )Math.pow(p, k) *
( float )Math.pow( 1 - p, n - k);
}
// Driver code
public static void main(String[] args)
{
int n = 10 ;
int k = 5 ;
float p = ( float ) 1.0 / 3 ;
float probability = binomialProbability(n, k, p);
System.out.print( "Probability of " +k);
System.out.print( " heads when a coin is tossed " +n);
System.out.println( " times where probability of each head is " +p);
System.out.println( " is = " + probability );
}
} /* This code is contributed by Mr. Somesh Awasthi */ |
# Python3 program to compute Binomial # Probability # function to calculate nCr i.e., # number of ways to choose r out # of n objects def nCr(n, r):
# Since nCr is same as nC(n-r)
# To decrease number of iterations
if (r > n / 2 ):
r = n - r;
answer = 1 ;
for i in range ( 1 , r + 1 ):
answer * = (n - r + i);
answer / = i;
return answer;
# function to calculate binomial r.v. # probability def binomialProbability(n, k, p):
return (nCr(n, k) * pow (p, k) * pow ( 1 - p, n - k));
# Driver code n = 10 ;
k = 5 ;
p = 1.0 / 3 ;
probability = binomialProbability(n, k, p);
print ( "Probability of" , k,
"heads when a coin is tossed" , end = " " );
print (n, "times where probability of each head is" ,
round (p, 6 ));
print ( "is = " , round (probability, 6 ));
# This code is contributed by mits |
// C# program to compute Binomial // Probability. using System;
class GFG {
// function to calculate nCr
// i.e., number of ways to
// choose r out of n objects
static int nCr( int n, int r)
{
// Since nCr is same as
// nC(n-r) To decrease
// number of iterations
if (r > n / 2)
r = n - r;
int answer = 1;
for ( int i = 1; i <= r; i++)
{
answer *= (n - r + i);
answer /= i;
}
return answer;
}
// function to calculate binomial
// r.v. probability
static float binomialProbability(
int n, int k, float p)
{
return nCr(n, k) *
( float )Math.Pow(p, k)
* ( float )Math.Pow(1 - p,
n - k);
}
// Driver code
public static void Main()
{
int n = 10;
int k = 5;
float p = ( float )1.0 / 3;
float probability =
binomialProbability(n, k, p);
Console.Write( "Probability of "
+ k);
Console.Write( " heads when a coin "
+ "is tossed " + n);
Console.Write( " times where "
+ "probability of each head is "
+ p);
Console.Write( " is = "
+ probability );
}
} // This code is contributed by nitin mittal. |
<?php // php program to compute Binomial // Probability // function to calculate nCr i.e., // number of ways to choose r out // of n objects function nCr( $n , $r )
{ // Since nCr is same as nC(n-r)
// To decrease number of iterations
if ( $r > $n / 2)
$r = $n - $r ;
$answer = 1;
for ( $i = 1; $i <= $r ; $i ++) {
$answer *= ( $n - $r + $i );
$answer /= $i ;
}
return $answer ;
} // function to calculate binomial r.v. // probability function binomialProbability( $n , $k , $p )
{ return nCr( $n , $k ) * pow( $p , $k ) *
pow(1 - $p , $n - $k );
} // Driver code $n = 10;
$k = 5;
$p = 1.0 / 3;
$probability =
binomialProbability( $n , $k , $p );
echo "Probability of " . $k ;
echo " heads when a coin is tossed "
. $n ;
echo " times where probability of "
. "each head is " . $p ;
echo " is = " . $probability ;
// This code is contributed by nitin mittal. ?> |
<script> // Javascript program to compute Binomial Probability // function to calculate nCr i.e., number of
// ways to choose r out of n objects
function nCr(n, r)
{
// Since nCr is same as nC(n-r)
// To decrease number of iterations
if (r > n / 2)
r = n - r;
let answer = 1;
for (let i = 1; i <= r; i++) {
answer *= (n - r + i);
answer /= i;
}
return answer;
}
// function to calculate binomial r.v. probability
function binomialProbability(n, k, p)
{
return nCr(n, k) * Math.pow(p, k) *
Math.pow(1 - p, n - k);
}
// driver program let n = 10;
let k = 5;
let p = 1.0 / 3;
let probability = binomialProbability(n, k, p);
document.write( "Probability of " +k);
document.write( " heads when a coin is tossed " +n);
document.write( " times where probability of each head is " +p);
document.write( " is = " + probability );
// This code is contributed by code_hunt.
</script> |
Output:
Probability of 5 heads when a coin is tossed 10 times where probability of each head is 0.333333 is = 0.136565
Reference :
stat200