Binomial Random Variables
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++
// 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
// 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
# 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#
// 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 // 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. ?> |
Javascript
<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
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