# Thread: a breakdown of the probability formula

1. ## a breakdown of the probability formula

Hi there,

I'm currently trying to wrap my head around the binomial distribution and I'm caught up on how the probability function works for calculating the probability of k successes in n observations.

P(X = k) = (n C k) p^k (1 - p)^(n-k)

I understand that the formula works, but I don't see why it works...mainly b/c I don't understand why the chance for failure is factored into the equation. Any help in explaining this to me would be appreciated.

jjmclell

2. Originally Posted by jjmclell
Hi there,

I'm currently trying to wrap my head around the binomial distribution and I'm caught up on how the probability function works for calculating the probability of k successes in n observations.

P(X = k) = (n C k) p^k (1 - p)^(n-k)

I understand that the formula works, but I don't see why it works...mainly b/c I don't understand why the chance for failure is factored into the equation. Any help in explaining this to me would be appreciated.

jjmclell
In $n$ trials the probability that the first $k$ are successes and the remaining $(n-k)$trials are failures is $p^k (1-p)^{n-k}$.

Now the probability of $k$ successes and $(n-k)$ failures in some other (specified order) is the same as this. So to get the probaility that we need ( $k$ successes and $(n-k)$ failures) irrespective or order we need to multiply by the number of distinct permutations of $n$ objects $k$ of one type and $n-k$ of the other where objects of the same type are indistinguishable. Which is where the binomial coefficient comes from.

RonL