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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

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According to the binomial theorem:

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

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

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