The Binomial Distribution

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The Binomial Distribution

The binomial distribution is a distribution which works for any set of events which has:

A fixed number of trials

Only two possible outcomes from each trial

The same probability for each outcome from each trial

Trials independent of each other

E.g. I flip a coin. It is either heads or tails. It cannot be both and it cannot be anything other than heads or tails. We can also say that I will achieve a result of heads or not heads. 

To make this into a binomial distribution, we need to know a few things. 

Number of trials = n

Number of successes  = r

Probability of success = p

Probability of failure = q

This leads us to the equation:

P(X=r) = (nCr)(p^r)(q^(n-r)) where X ~ B(n, p)

x ~ B(n, p) simply means X is distributed binomially with number of trials, n, and probability of success, p.

If you have a graphical calculator which has a statistics setting, go to that and then find:

Dist (Distribution)

Binm (Binomial)

BcD (which gives you X=r)


BpD (which gives Xr)

Mean and Variance

The mean and variance of the binomial distribution can be found easily using nothing more than n and p.

Mean = np

Variance = np(1-p)

The standard deviation of the binomial distribution can be found by taking the square root of the variance.

These two can be found in the formula book on


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