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 ) = Pr ⁡ ( K = k ) = ( n 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 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 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.

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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  )  =  Pr  ⁡  (  K  =  k  )  =                              (                            n        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 &quot;                              (                            n        k                              )                    &quot; that we multiply at the start of the formula is a positive integer, and I am confused that why shouldn&#039;t the net product be more than one?I mean, the fraction that we get after multiplying the probabilities, wouldn&#039;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 &quot;                              (                            n        k                              )                    &quot; ?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&#039;t see why it shouldn&#039;t be greater than 1. I am in learner&#039;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.

coquinarq1 · Accepted Answer

According to the binomial theorem:  1  =      1    n    =  (  p  +  1  −  p      )    n    =      ∑          k      =      0        n                                (                            n        k                              )                          p    k    (  1  −  p      )          n      −      k

I have finished elementary probability and I know the sum of all probabilites in a data set is 1.But

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