Binomial distribution

Flip a coin n times.

Define R. V. $X_i$ to be the results of i-th flip (see above sect. ).

Summation of R. V.'s are also an R.V., and let's call this new R.V. as Y:
$Y = X_1 + X_2 + ... + X_n$

The meaning of this new R.V. is the number of heads after n coin flipping, so the possible values for Y are ${0, 1, \cdots, n}$.

• Calculate the probability that Y = 0 with multiplication rule:
  
  
  
  
  
  
  
  

• Calculate Pr[Y = 1] with multiplication and addition rules:
  
  
  
  
  
  
  
  

• Calculate Pr[Y = 2] with multiplication and addition rules:
  
  
  
  
  
  
  
  

• Generalize to Pr[Y=k]