# Calculating Binomial Probabilities

The Normal Approximation of the Binomial Distribution
14:58 — by Daniel Schaben

Tip: This isn't the place to ask a question because the teacher can't reply.

## 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:

1. The experiment consists of n repeated trials.
2. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
3. The probability of success, denoted by p, is the same on every trial.
4. 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
$Y = \Sigma \left({X}_{i}\right)$

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(1-p))

PAY ATTENTION: in binomial the approx is good when $n > 20$, $n p > 5$ and $n \left(1 - p\right) > 5$

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