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 : a busy cat

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 so please go easy on me.(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 depending solely on luck!).Here's what the description of the formula says :

Probability mass function

In general, if the random variable K follows the binomial distribution with parameters n and p, we write K ~ B(n, p). The probability of getting exactly k successes in n trials is given by the probability mass function: a busy cat

for k = 0, 1, 2, ..., n, where

a busy cat

PS:I sourced the above formula from the following link

Link for the formula and its description


According to the binomial theorem:

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

  • $\begingroup$ You nailed it in a single clear line.Thanks!! $\endgroup$ – Ivy Mike Dec 8 '12 at 12:28
  • 1
    $\begingroup$ Mind blown. Brilliant connection between the two. Thank you so much $\endgroup$ – theideasmith Mar 22 '17 at 2:09

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