Binomial(Beta(a,b,1),n)

A Beta-Binomial distribution returns a discrete value between 0 and n. An example of a Beta-Binomial(30,10,7) distribution, together with its best matching Binomial distribution is given below:

### Uses

The Beta-Binomial distribution is used to model the number of successes in n binomial trials (usually =Binomial(p,n)) but when the probability of success p is also a random variable, and can be adequately described by a Beta(*a**,b,1*) distribution. In other words:

Beta-Binomial(*n*,*a**, b*) =Binomial(Beta(*a**,b,1*), *n*) -

The extreme flexibility of the shape of the Beta distribution means that it is often a very fair representation of the randomness of p.

The probability of success varies randomly, but in any one scenario that probability applies to all trials. For example, you might consider using the Beta-Binomial distribution to model:

The number of cars that crash in a race of n cars, where the predominant factor is not the skill of the individual driver, but the weather on the day;

The number of bottles of wine from a producer that are bad where the predominant factor is not how each bottle is treated, but something to do with the batch as a whole;

The number of people who get ill at a wedding from n invited, all having a taste of the delicious souffle, unfortunately made with contaminated eggs, where their risk is dominated not by their individual immune system, or the amount they eat, but by the level of contamination of the shared meal.

### Generation

The Beta-binomial distribution is not directly available in Crystal Ball but can be constructed by nesting a Beta distribution inside a Binomial distribution as is shown in the model Beta-Binomial. This method loses some of the advantage of Latin Hypercube sampling, as it is sampling from two distributions to create one, which can be avoided if you construct the distribution.

The links to the Beta-Binomial software specific models are provided here:

### Comments

The Beta-Binomial distribution is always more spread than its best fitting Binomial distribution, because the Beta distribution adds extra randomness. Thus, when a Binomial distribution does not match observations, because the observations exhibit too much spread, a Beta-Binomial distribution is often used instead.