Main Concept

The binomial distribution is a discrete probability distribution that is used to obtain the probability of observing exactly k number of successes in a sequence of n trials, with the probability of success for all single trials of p.

The shape of the binomial distribution is dependent on the values of n and p, as illustrated in the following diagrams:

The normal distribution is a continuous probability distribution whose shape is dependent on the mean, m, and variance, σ.

Since a binomial variate, B(n,p), is a sum of n independent, identically distributed Bernoulli variables with parameter p, it follows that by the central limit theorem it can be approximated by the normal distribution with mean $n p$ and variance $n p \left(1-p\right)$, provided that both $n p \>5$ and $n \left(1-p\right) \> 5$.

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