Binomial Probability Distributions with Microsoft Excel

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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 called a success, then the random variable
x would be the number of heads observed in n flips. 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 still 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 allowable risk (alpha) for rejecting an acceptable lot.

The CRITBINOM function uses the following 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 will 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 will 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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