In questo post parleremo delle variabili casuali binomiali.
Prerequisito: Variabili casuali 
Un tipo specifico di discreto variabile casuale che conta la frequenza con cui si verifica un particolare evento in un numero fisso di tentativi o prove. 
Affinché una variabile sia una variabile casuale binomiale devono essere soddisfatte TUTTE le seguenti condizioni: 
 

1. Esiste un numero fisso di prove (una dimensione del campione fissa).
2. Ad ogni prova l'evento di interesse si verifica oppure no.
3. La probabilità che si verifichi (o meno) è la stessa in ciascuna prova.
4. Le prove sono indipendenti l'una dall'altra.

Notazioni matematiche 
 

 n = number of trials   
 p = probability of success in each trial   
 k = number of success in n trials  
  
 Cerchiamo ora di determinare la probabilità di k successo in n prove.   
 Qui la probabilità di successo in ciascuna prova è p indipendente dalle altre prove.    
 Quindi scegliamo prima k prove in cui ci sarà un successo e nelle restanti n-k prove ci sarà un fallimento. Ci sono molti modi per farlo    
    
   
  
 Poiché tutti gli n eventi sono indipendenti, la probabilità di k successo in n prove è equivalente alla moltiplicazione della probabilità per ciascuna prova.   
 Qui ci sono k successi e nk fallimenti. Quindi la probabilità per ogni modo di ottenere k successi e nk fallimenti è    
    
   
  
 Quindi la probabilità finale è    
    
 
 (number of ways to achieve k success   
  and n-k failures)   
  *   
 (probability for each way to achieve k   
  success and n-k failure)  
  
 Allora la probabilità della variabile casuale binomiale è data da:    
    
   
  
 Sia X una variabile casuale binomiale con il numero di prove n e la probabilità di successo in ciascuna prova pari a p.    
 Il numero di successi previsto è dato da    
    
 
 E[X] = np  
  
 La varianza del numero di successi è data da    
    
 
 Var[X] = np(1-p)  
  
    Esempio 1    : Considera un esperimento casuale in cui una moneta distorta (probabilità di testa = 1/3) viene lanciata per 10 volte. Trova la probabilità che il numero di teste che escono siano 5.   
 Soluzione:    
    
 
 Let X be binomial random variable    
 with n = 10 and p = 1/3   
 P(X=5) = ?    
      
    
 Ecco l'implementazione dello stesso    
    
 C++   
   // C++ program to compute Binomial Probability   #include          #include         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.   
  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>   
  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.   ?>   
   
    Produzione:        
    
 
 Probability of 5 heads when a coin is tossed 10 times where probability of each head is 0.333333   
  is = 0.136565  
  
    
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