Understanding the binomial probability coefficient and factorial notation.

What is a binomial coefficient and factorial notation?

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...= y^3 +3xy^2+3x^2y+x^3.

the numbers (coefficients) in each of the 4 terms of this polynomial are the binomial coefficients (1,3,3,1).

You are probably more familiar with the x and y being replaced by p and 1-p. the binomial distribution looks like...

(n choose k) p^k (1-p)^n-k

where p is the probability of "success" and (1-p) is the probability of "failure".

So, lets see an example...

We want to find the probability of rolling at least 1 six in 4 rolls of a single die.
A success is seen as rolling a 6, this has probability equal to 1 sixth. Let us define a random variable

X = total number of 6's in 4 rolls

So, X has a binomial distribution, right. The roll can be a success (a 6) or a failure (not a 6). We have n=4 (4 rolls), p=1/6 (chance of getting a 6)

Now lets find this probability....

I'd like you to see that ...