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Binomial Distribution

Binomial distribution is a discrete probability distribution. Binomial distribution estimates probability distribution when there are "n" number of independent trials resulting in either success or failure. Following are the conditions for binomial distribution :

• There are "n" number of trails, where "n" is a finite number.
• These "n" trails are all mutually exclusive.
• There can only be two possible outcomes in every trial - "success" or "failure", "yes" or "no".
• Here, the probability of success is constant which is "p" for each trial.
• The probability of failure in each trials also a constant, say q. Eventually, q = 1 - p.

Formula for Binomial Distribution
The binomial probability function obtaining "k" successes in "n" trails is given by:

$B(k;n,p) =\ _{k}^{n}\textrm{C}\ p^{k}\ (1-p)^{n-k}$Where,
n = Total number of trials.
p = Probability of success in each trail.
1 - p = q = Probability of failure in each trial.
k = Number of successes in total n trials.

$_{k}^{n}\textrm{C}$ is known as binomial coefficient and is also written as C(n , k) or $\binom{n}{k}$.

Binomial distribution is known as Bernoulli distribution, when n = 1.If n trials are carried out with replacement, then the trials are independent. In this case, binomial distribution is used. It is very accurate and widely used to model "k" number of successes in "n" mutually exclusive trials. If experiment is done without replacement, then trials are not independent. In this case, binomial distribution can not be used.

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