# The state of individuals sampled from a large or infinite population

Scenario 1: There are two possible states for each individual

If samples are taken from a very large or infinite (e.g. constantly regenerating) population, then the probability that a sample will be in a particular state (e.g. infected, faulty, going to vote Labour, male, over two meters tall, etc.) is simply the prevalence of that characteristic in the population. In that case, the sampling is a binomial process, and the number of individuals s in a sample of size n that will have some particular characteristic, where the prevalence of that characteristic is p, is given by:

s = Binomial(p,n)

Examples:

• 20% of the bulls of country X have disease Y. If 35 bulls are taken from that population, how many will be infected with Y? Answer:  Binomial(20%,35)

• A manufacturer produces AC adapters for laptops. If there is a 2% chance that an adapter is faulty, how many faulty adapters will there be in a consignment of 100?

•  Advertising brochures posted to households produce a 0.3% response rate. If 100,000 brochures are sent out, how many responses will there be?

Scenario 2: There are several possible states for each individual

Sometimes we are interested in knowing which of several mutually exclusive (i.e. cannot occur together) and exhaustive (i.e. include all possible outcomes) states individuals are taking in a random sample from a population. In this case, the sampling is a multinomial process, and the number of individuals in the sample that take each possible state is given by a multinomial distribution Multinomial(n, {p}). Crystal Ball does not offer such a distribution, which is an array calculation (i.e. it takes up several cells in a spreadsheet), but a multinomial distribution can easily be constructed as a series of sequential binomial distributions. Model Multinomial provides an example. An explanation of the reasoning behind the model's construction is given in the section on multivariate trials.

The links to the Multinomial software specific models are provided here:

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