Binomial Probability Distribution
Binomial Probability Distribution
A binomial distribution describes the outcome of a multi-step experiment, consisting of n
identical trials, where each trial ends in either a success or a failure and the probability of a
success p does not change from trial to trial.
This useful statistical analysis can be performed relatively easily using Microsoft Excel using
the Excel BINOMDIST, CRITBINOM and NEGBINOMDIST functions.
Note, however, that when making binomial probability calculations, the trials must also be
independent so that success in one trial does not affect the probability of success in another
trial. The binomial random variable x is the number of successes observed in n trials.
If samples are not replaced, and therefore the outcome of one trial changes the probability of
success in another trial, you need to use the hypergeometric probability distribution Excel
function.
Using Excel's BINOMDIST Function
For example, if you flip a coin n times and "heads" is cal ed a success, then the random
variable x would be the number of heads observed in n flips.
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It could take the values 1,2,3,...,n with different probabilities. The BINOMDIST function uses
the following syntax:
=BINOMDIST(x,n,p,cumulative)
If you want to find the probability of exactly x successes, enter FALSE as the fourth
(cumulative) argument. If you want to find the probability of x or fewer successes, enter TRUE
as the fourth argument.
For example, if you were to flip a fair coin 20 times and wanted to find the probability of it
turning up "heads" exactly 10 times, the function looks like this:
=BINOMDIST(10,20,0.5,FALSE)
The function returns the value 0.176197052. If you wanted to find the probability of getting 10
or fewer heads, you replace the FALSE with TRUE, and the function returns the value
0.588098526.
Using Excel's CRITBINOM Function
The acceptance criterion function, CRITBINOM, is used for quality control of a production
process. You use this function to find the maximum number of defective items that a person
can find in a lot and stil allow acceptance of the lot.
Inspectors should accept the lot if they find this number or fewer defective items and reject the
lot if they find more defective items.
To determine the acceptance criterion, you need to know the number of items in the lot, the
probability of accepting each item, and the producer's al owable risk (alpha) for rejecting an
acceptable lot.
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The CRITBINOM function uses the fol owing syntax:
=CRITBINOM (trials, probability_s, alpha)
where trials is the number of trials, probability's is the probability of a success on each trial,
and alpha is the criterion value. Probability's and alpha are both between 0 and 1.
Using Excel's NEGBINOMDIST Function
If the number of successes is fixed in a binomial distribution and you want to find the number
of trials, use the NEGBINOMDIST function. This function returns the probability that there wil
be a certain number of failures before the threshold number of successes, given the constant
probability of a success.
For example, if you need to find 20 straight 2 by 4s from a stack, and you know the probability
that a board in the stack is straight is 0.2 (20%), you can use the NEGBINOMDIST to find that
there is about a 2% probability that you wil reject 75 boards before finding all 20 straight ones.
The NEGBINOMDIST function uses the following syntax:
=NEGBINOMDIST (number failures, number successes, probability of success)
For this example, the function looks like this:
=negbinomdist (75, 20, 0.2)


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