8
$\begingroup$

I have finished elementary probability and I know the sum of all probabilites in a data set is 1.But while reading Binomial Distribution,I encountered the formula for the Probability mass distribution :

$$f(k ; n, p)=\operatorname{Pr}(K=k)= {n \choose k} p^{k}(1-p)^{n-k}$$

Well, I know that probability is a fraction less than $1$, and fractions multiplied with fractions will still yield a lesser fraction as the product. But what I am confused about is that the "${n \choose k}$" that we multiply at the start of the formula is a positive integer, and I am confused that why shouldn't the net product be more than one?

I mean, the fraction that we get after multiplying the probabilities, wouldn't that turn greater than $1$ if we multiply (which is repeated addition by itself) that fraction by the positive integer we get as a result of "${n \choose k}$" ?

Sorry about my roundabout way of talking. You are only requested to prove that the whole thing is a value less than $1$. I tried and just couldn't see why it shouldn't be greater than $1$. I am in learner's stage (I could have learnt the forumla by rote, but my video instructor says if I proceed like that without understanding things and asking questions, then I will be like a monkey on a type-writer)

Here's the wikipedia page about it

$\endgroup$

1 Answer 1

28
$\begingroup$

According to the binomial theorem:

$$ 1 = 1^n = (p+1-p)^n = \sum_{k=0}^n {n \choose k} p^k(1-p)^{n-k} $$

$\endgroup$
2
  • $\begingroup$ You nailed it in a single clear line.Thanks!! $\endgroup$
    – Ivy Mike
    Dec 8, 2012 at 12:28
  • 1
    $\begingroup$ Mind blown. Brilliant connection between the two. Thank you so much $\endgroup$ Mar 22, 2017 at 2:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.