Calculating Binomial Probabilities
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Key Questions

Binomial Formula.
Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success on an individual trial is P, then the binomial probability is:
#b(x; n, P) = nCx * p^x * q^(n  x) # ; q = 1 p( A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties:
 The experiment consists of n repeated trials.
 Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
 The probability of success, denoted by p, is the same on every trial.
 The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials. )
Example:
Suppose a die is tossed 5 times. What is the probability of getting exactly 2 fours?
Solution:
This is a binomial experiment in which the number of trials is equal to 5, the number of successes is equal to 2, and the probability of success on a single trial is 1/6 or about 0.167. Therefore, the binomial probability is:#b(2; 5, 0.167) = 5C2 * (0.167)^2 * (0.833)^3 = 0.161# 
Do you mean "n choose x"
if x = 1 then a Bernoulli experiment was conducted.
if x > 1 then multiple Bernoulli experiments were conducted.
The calculation of this piece is essential to computing the correct binomial probabilities.

Because of Central limit theorem.
Central limit theorem says that if you have n random variables iid with same mean and std deviation (we call
#mu# and#sigma# ), and we study Y dafined as
#Y = Sigma(X_i)# As n approaches infinity, Y converge in distribution at a normal distribution with parameter (
#mu_Y = n mu, sigma_Y = sigma sqrt(n) # ).So binomial is the sum of n iid Bernoulli distribution, so CLT is valid and we can say that
#Y ~ Bin(n,p) >^d Norm(np,np(1p))# PAY ATTENTION: in binomial the approx is good when
#n>20# ,#n p >5# and#n (1p) >5#