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**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

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**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 )__

*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) =
^{n}C_{x}q^{(n-x)}p^{x}, where q = 1 - p

p can be considered as the probability of a success, and q the probability of a failure.

Note:

^{n}C_{r}(“n choose r”) is more commonly written , 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).

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**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.