How is a binomial distribution different from a Bernoulli distribution?

1 Answer
Nov 22, 2016

the Binomial is just the aggregate of many Bernoulli trials

Explanation:

if the Bernoulli is #f(x) = p(x)1-p(x)# and we wanted to know what would happen if we had two trails then there are 4 total possibilities that can happen.

1= success , 2= success: #p(x)^2(1-p(x))^0#
1= fail , 2= success: #p(x)(1-p(x))#
1= success , 2= fail: #p(x)(1-p(x))#
1= fail , 2= fail: #p(x)^0(1-p(x))^2#

now if we wanted to know the probability of exactly 1 success out of 2 trials we know that

#2*p(x)(1-p(x))# or #(""_1^2)p(x)^1(1-p(x))^(2-1)#

if we want exactly 2 success out of 2 trials then

#p(x)^2(1-p(x))^0# or #(""_2^2)p(x)^2(1-p(x))^(2-2)#

it turns out that if you continue like this you end up with the binomial form
#f(x,k,n) = (""_k^n)p(x)^k(1-p(x))^(n-k)#