Binomial Distribution :


Binomial distribution

gives us the probabilities of interests are those of receiving a certain number of success r in n independent trails each having two possible outcomes and sample probability p , of success so for example using a binomial distribution we can determine the probability of giving h heads into coin toss coin

How does the binomial distribution do this?

Basically a two part process is involved, first we have to determine the

probability of one possible way the event can occur and then determine the number of different way the event can occur that is P(Event) = (number of ways event can occur * p(one occurrence )

if a discrete random variable X has the following probability density function (p.d.f.), it is said to have a binomial distribution:
  • P(X = x) = nCx q(n-x)px, where q = 1 - p
p can be considered as the probability of a success, and q the probability of a failure.
Note: nCr (“n choose r”) is more commonly written clip_image001, but I shall use the former because it is easier to write on a computer. It means the number of ways of choosing r objects from a collection of n objects (see permutations and combinations).
If a random variable X has a binomial distribution, we write X ~ B(n, p) (~ means ‘has distribution…’).
(n) and (p) are known as the parameters of the distribution (n can be any integer greater than 0 and p can be any number between 0 and 1). All random variables with a binomial distribution have the above p.d.f., but may have different parameters (different values for n and p).

What is the relation between the binomial and normal distributions?

When the number of trials is large and when the probability of success is not extreme (i.e., neither close to 0 nor close to 1), then the normal distribution may be used to very closely approximate results from the binomial distribution.
Note: When the number of trials is greater than 1,000, the Binomial Calculator uses a normal distribution to estimate the binomial probabilities. In most cases, this yields good results - often accurate within 1 or 2 percent. The larger the number of trials, the more accurately the normal distribution estimates binomial probabilities.