Question

How is a binomial distribution different from a Bernoulli distribution?

Answer 1

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)`