I dette innlegget vil vi diskutere binomiale tilfeldige variabler.
Forutsetning: Tilfeldige variabler 
En bestemt type diskret tilfeldig variabel som teller hvor ofte en bestemt hendelse inntreffer i et fast antall forsøk eller forsøk. 
For at en variabel skal være en binomial tilfeldig variabel, må ALLE følgende betingelser være oppfylt: 
 

1. Det er et fast antall forsøk (en fast prøvestørrelse).
2. Ved hver rettssak inntreffer eller ikke hendelsen av interesse.
3. Sannsynligheten for forekomst (eller ikke) er den samme for hver rettssak.
4. Prøver er uavhengige av hverandre.

Matematiske notasjoner 
 

 n = number of trials   
 p = probability of success in each trial   
 k = number of success in n trials  
  
 Nå prøver vi å finne ut sannsynligheten for k suksess i n forsøk.   
 Her er sannsynligheten for suksess i hvert forsøk p uavhengig av andre forsøk.    
 Så vi velger først k-forsøk der det vil være en suksess og i hvile-n-k-forsøk vil det være en fiasko. Antall måter å gjøre det på er    
    
   
  
 Siden alle n hendelser er uavhengige, er sannsynligheten for k suksess i n forsøk ekvivalent med multiplikasjon av sannsynlighet for hver prøvelse.   
 Her er det k suksess og n-k fiaskoer, så sannsynligheten for hver måte å oppnå k suksess og n-k fiasko er    
    
   
  
 Derfor er endelig sannsynlighet    
    
 
 (number of ways to achieve k success   
  and n-k failures)   
  *   
 (probability for each way to achieve k   
  success and n-k failure)  
  
 Da er binomial tilfeldig variabel sannsynlighet gitt av:    
    
   
  
 La X være en binomial tilfeldig variabel med antall forsøk n og sannsynligheten for å lykkes i hvert forsøk være p.    
 Forventet antall suksess er gitt av    
    
 
 E[X] = np  
  
 Varians av antall suksess er gitt av    
    
 
 Var[X] = np(1-p)  
  
    Eksempel 1    : Tenk på et tilfeldig eksperiment der en skjev mynt (sannsynlighet for hode = 1/3) kastes 10 ganger. Finn sannsynligheten for at antallet hoder som vises vil være 5.   
 Løsning:    
    
 
 Let X be binomial random variable    
 with n = 10 and p = 1/3   
 P(X=5) = ?    
      
    
 Her er implementeringen for det samme    
    
 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.   ?>   
   
    Produksjon:        
    
 
 Probability of 5 heads when a coin is tossed 10 times where probability of each head is 0.333333   
  is = 0.136565  
  
    
         Lag quiz