Derivation of the probability distribution function of a binomial distribution

[misogynisticfeminist]

Is there a way to derive

$$P (X=r) =^nC_r p^r q^{n-r} , r= 0, 1, 2,..., n$$

where $$X: B(n,p)$$

where n is the total number of bernoulli experiments,

p the probability of success

q, the probability of failure.

 

[matt grime]

Yes, just think about it, it's just simple combinatorics (you omitted to mention independent trials, which is, I'm sure important), and no harder than working out how to choose r from n (in fact it is the same).

 

[robert Ihnot]

This is not a problem as matt goes into. Suppose we have XXX and YY,then how many ways can combinations occur? Well there are five elements in 5! ways, but 3 of them are similar and the other two are similar, so it's 5!/(3!2!)=10 distinct ways. YYXXX, YXYXX, YXXYX, YXXXY, XYXXY, XXYXY, XXXYY, XYYXX, XXYYX, XYXYX.