Understanding the binomial probability coefficient and factorial notation.
What is a binomial coefficient and factorial notation?© SolutionLibrary Inc. solutionlibary.com 9836dcf9d7 https://solutionlibrary.com/statistics/probability/understanding-the-binomial-probability-coefficient-and-factorial-notation-1f3
...= 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 ...